{"id":33949,"date":"2021-05-09T13:00:27","date_gmt":"2021-05-09T13:00:27","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33949"},"modified":"2025-08-01T04:54:18","modified_gmt":"2025-08-01T04:54:18","slug":"dominia-integritatis-et-numeri-integri","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/dominia-integritatis-et-numeri-integri\/","title":{"rendered":"Dom\u00ednia Integrit\u00e1tis et Numeri Integri"},"content":{"rendered":"<style>\n\tp, ul, ol{\n\ttext-align: justify;\n\t}\n\th1{\n\ttext-align:center;\n\ttext-transform: uppercase;\n\t}\n\th2{\n\ttext-align:center;\n\ttext-transform: uppercase;\n\tfont-size:24pt;\n\t}\n\th3 { \n\t\ttext-align: center;\n\t\ttext-transform: uppercase;\n\t\tfont-size: 24px !important;\n\t}\n<\/style>\n<h1>Dom\u00ednia Integrit\u00e1tis et Numeri Integri<\/h1>\n<p style=\"text-align:center;\"><em><strong>Summarium:<\/strong><br \/>\nHac in lectione notio Dominii Integritatis introducitur, eiusque momentum in studio algebrae generalis explicatur, et nonnullae proprietates eius maximi momenti per probationes formales demonstrantur. <\/em><\/p>\n<p style=\"text-align:center;\"><em><strong>Metas Discendi:<\/strong><\/em><br \/>\nHac lectione peracta, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Intellegere<\/strong> finem studii algebrae generalis.<\/li>\n<li><strong>Intellegere<\/strong> notionem dominii integritatis.<\/li>\n<li><strong>Explicare<\/strong> elementa fundamentalia communia inter dom\u00ednia integrit\u00e1tis et n\u00fameros \u00edntegros.<\/li>\n<li><strong>Demonstr\u0101re<\/strong> per probationes formales proprietates fundamentales dominiorum integritatis.<\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<strong><u>INDEX CONTENTORUM<\/u><\/strong><br \/>\n<a href=\"#1\">FINIS ALGEBRAE GENERALIS ET PRAENOTIONES<\/a><br \/>\n<a href=\"#2\">A NUMERIS INTEGRIS AD DOMINIA INTEGRITATIS<\/a><br \/>\n<a href=\"#3\">ELEMENTA COMMUNIA INTER DOMINIA INTEGRITATIS ET NUMEROS INTEGROS<\/a><br \/>\n<a href=\"#4\">PROPRIETATES DOMINIORUM INTEGRITATIS ET NUMERORUM INTEGRORUM<\/a><br \/>\n<a href=\"#5\">EXERCITATIONES<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/hxmc1-eXWxU?si=57GADT52JG4fHFT-\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Finis Algebrae Generalis et Praenotiones<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=183s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Finis principalis algebrae generalis<\/span><\/strong><\/a> est studium totius varietatis systematum mathematicorum possibilis. Hic plura talia systemata investigabimus, inter quae praecipua eminent numeri naturales et integri, per quos ad dominia integritatis perveniemus.<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{N}= \\{1,2,3,4,\\cdots\\}<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}= \\{0,\\pm 1,\\pm 2,\\pm 3,\\pm 4,\\cdots\\}<\/span><\/span><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>A Numeris Integris ad Dom\u00ednia Integrit\u00e1tis<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=358s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Examen nostrum incipiemus a numeris integris,<\/span><\/strong><\/a> et ratio cur hoc modo procedamus est quod hi numeros plurimas similitudines cum plurimis systematibus numericis quae in hoc studio examinabimus exhibent.<\/p>\n<p>Pro eo quod definitionem numerorum integrorum statuere conemur, initium capiemus supponentes eos, quidquid sint, quasdam proprietates satisfacere. Ad hoc seligitur systema axiomatum, quo omnes proprietates quas intuitu ad numeros integros referre solemus demonstrari possint.<\/p>\n<p>Haec omnia fiunt per <strong>Peano Axiomata<\/strong> Numerorum Naturalium, dum operationes arithmeticae fundamentales introducuntur. Hoc modo axiom\u00e1tico utens et operationes diversas super naturales atque integros extendens, nova genera numerorum obtinentur, ut puta rationales, irrationales, reales, complexi, quaternionici, octonionici, et plura alia.<\/p>\n<p>Postea, si numeros integros observemus, videbimus eos proprietates possidere quae in omnibus fere numerorum generibus repetuntur, ut existentia elementi neutri multiplicativi, neutri additivi, et leges distributivae. Quare de his rebus loquentes, sermonem constituere possumus qui nobis permittit de omnibus illis simul tractare. In hoc contextu emergunt vocabula ut<\/p>\n<ul>\n<li>Dominium Integritatis<\/li>\n<li>Annulus<\/li>\n<li>Coetus<\/li>\n<li>Spatium Vectoriale<\/li>\n<\/ul>\n<p>Et multa alia huiusmodi vocabula. Nos autem nostras vires in studio <strong>Dominiorum Integritatis<\/strong> imprimis collocabimus.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Elementa Fundamentalia Communia Inter Dom\u00ednia Integrit\u00e1tis et N\u00fameros \u00cdntegros<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=472s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ut explicemus quid sit dominium integritatis<\/span><\/strong><\/a>, utamur proprietatibus quae ex numeris integris bene cognoscuntur. In hoc contextu, si <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>, et <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> sint numeri integri, tunc valent <strong>leges<\/strong><\/p>\n<ol>\n<li><strong>Commutativae:<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = b + a<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab = ba<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong>Associativae:<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+(b+c) = a+b+c = (a+b)+c<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(ab)c = abc = a(bc)<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong>Distributivae:<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+(b+c) = a(b+c) = ab+ac<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>Praeterea, adsunt quaedam elementa specialia, scilicet neutra:<\/p>\n<ul>\n<li><strong>Neutrale Additivum:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = a \\leftrightarrow c=0<\/span><\/span><\/li>\n<li><strong>Neutrale Multiplicativum:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac = a \\leftrightarrow c=1<\/span><\/span><\/li>\n<\/ul>\n<p>Res cuius symbolum est <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> est neutrum adductionis; cui symbolum est <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> est neutrum multiplicationis.<\/p>\n<p>Numeri integri etiam possident inversa additiva. Cuique numero integro respondet suum inversum additivum, quod cum ipso additum dat neutrum additivum.<\/p>\n<ul>\n<li><strong>Inversum Additivum:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = 0 \\longleftrightarrow c=-a<\/span><\/span><\/li>\n<\/ul>\n<p>Inversa additiva per signum \u00ab-\u00bb designantur quod eis adest.<\/p>\n<p>Denique adest <strong>lex simplificationis<\/strong>, quae per relationem sequens exprimitur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(c\\neq 0 \\wedge ca = cb) \\longleftrightarrow (a=b)<\/span><\/span><\/p>\n<p>Hae proprietates quas recensuimus valent etiam in multis aliis systematibus: realibus, complexis, polynomialibus, etc. Quamobrem <strong>Dominium Integritatis<\/strong> vocamus omnem collectionem quae his proprietatibus satisfacit.<\/p>\n<p><span style=\"color: #800000;\"><strong>DEFINITIO:<\/strong><\/span> Dominium integritatis est quaelibet collectio <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span>, quae operationibus additionis et multiplicationis instructa est ita ut<\/p>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in D \\longrightarrow a+b \\in D<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in D \\longrightarrow ab \\in D<\/span><\/span><\/li>\n<\/ul>\n<p>Praeterea satisfiunt leges <strong>associativae, commutativae<\/strong> et <strong>distributivae,<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> continet <strong>neutra additiva<\/strong> et <strong>multiplicativa<\/strong> (quorum utrumque unicum est), et denique valet <strong>lex simplificationis.<\/strong><\/p>\n<h4>Exemplum Dominii Integritatis<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=749s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Consideremus collectionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\{a+b\\sqrt{3}\\; |\\; a,b\\in \\mathbb{Z}\\}.<\/span><\/span><\/span> <\/strong><\/a>Haec collectio, operationibus additionis et multiplicationis ordinariis instructa, est dominium integritatis quia leges commutativitatis, associativitatis et distributionis complet; habet neutra additiva et multiplicativa necnon etiam inversum additivum.<\/p>\n<ul>\n<li><strong>Neutrale Additivum:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>Neutrale Multiplicativum:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>Inversum Additivum:<\/strong> Omni elemento <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b\\sqrt{3}<\/span><\/span> respondet inversum additivum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-a-b\\sqrt{3}<\/span><\/span><\/li>\n<\/ul>\n<p>Et hoc maxime est momenti: Haec collectio A clausa est sub operationibus additionis et multiplicationis, id est, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> tum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x+y\\in A<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xy\\in A.<\/span><\/span> Quod facile comprobandum est: Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_1 + b_1\\sqrt{3}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_2 + b_2\\sqrt{3}<\/span><\/span> sunt elementa <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>, tum habebimus<\/p>\n<p style=\"text.align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(a_1 + b_1\\sqrt{3}) + (a_2 + b_2\\sqrt{3}) &amp;=(a_1+a_2) + (b_1 + b_2)\\sqrt{3} \\in A\\\\ \\\\\n\n(a_1 + b_1\\sqrt{3})  (a_2 + b_2\\sqrt{3})  &amp;= a_1a_2 + a_1b_2\\sqrt{3}+b_1a_2\\sqrt{3} + 3b_1b_2 \\\\\n\n&amp;=(a_1a_2 + 3b_1b_2) + (a_1b_2 + b_1a_2)\\sqrt{3} \\in A\n\n\\end{array}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/y6jXNPhjKv4?si=9SaXhWHN42sC73lZ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Proprietates Dominiorum Integritatis et Numerorum Integrorum<\/h3>\n<h4>Neutrale Additivum Dominii Integritatis Unicum Est<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=26s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Hoc demonstrari potest per reductionem ad absurdum:<\/span><\/strong><\/a> Fingamus duo neutra additiva existere, sint <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0^\\prime<\/span><\/span> talia elementa. Tum habebimus:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; 0\\neq 0^\\prime &amp; \\text{; Praemissa}\\\\\n\n(2) &amp; a+0 = a &amp; \\text{; Praemissa: $0$ est neutrum additivum}\\\\\n\n(3) &amp; b+0^\\prime = b &amp; \\text{; Praemissa: $0^\\prime$ est neutrum additivum}\\\\\n\n(4) &amp; 0^\\prime + 0 = 0^\\prime &amp; \\text{; Substituendo $a=0^\\prime$ in $(2)$}\\\\\n\n(5) &amp; 0 + 0^\\prime = 0 &amp; \\text{; Substituendo $b=0$ in $(3)$}\\\\\n\n(6) &amp; 0 = 0^\\prime  &amp; \\text{; Ex $(4,5)$ et commutativitate additionis}\\\\\n\n(7) &amp; \\bot &amp;\\text{; Ex $(1,6)$}\n\n\\end{array}<\/span>\n<p>Ex hoc ratiocinio concludimus igitur:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0 \\neq 0^\\prime, a + 0 = a, b + 0^\\prime = b\\}\\vdash \\bot.<\/span><\/span><\/p>\n<p>Ergo, per reductionem ad absurdum habetur<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{a + 0 = a, b + 0^\\prime = b\\}\\vdash 0 = 0^\\prime.<\/span><\/span><\/p>\n<p>Id est, si duo neutra additiva adsint, tunc idem sunt, atque ideo est unicum.<\/p>\n<h4>Neutrale Multiplicativum Etiam Unicum Est<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=305s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Demonstratio fere est omnino similis priori.<\/span><\/strong><\/a> Si duo adsint: <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1^\\prime<\/span><\/span>, tunc sequens argumentatio fieri potest:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; 1\\neq 1^\\prime &amp; \\text{; Praemissa}\\\\\n\n(2) &amp; 1\\cdot a = a &amp; \\text{; Praemissa: $1$ est neutrum multiplicativum}\\\\\n\n(3) &amp; 1^\\prime \\cdot b = b &amp; \\text{; Praemissa: $1^\\prime$ est neutrum multiplicativum}\\\\\n\n(4) &amp; 1\\cdot 1^\\prime = 1^\\prime &amp; \\text{; Substituendo $a=1^\\prime$ in $(2)$}\\\\\n\n(5) &amp; 1^\\prime \\cdot 1 = 1 &amp; \\text{; Substituendo $b=1$ in $(3)$}\\\\\n\n(6) &amp; 1 = 1^\\prime  &amp; \\text{; Ex $(4,5)$ et commutativitate}\\\\\n\n(7) &amp; \\bot &amp;\\text{; Ex $(1,6)$}\n\n\\end{array}<\/span>\n<p>Ex quo concluditur igitur:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{1 \\neq 1^\\prime, 1a= a, 1b = b\\}\\vdash \\bot.<\/span><\/span><\/p>\n<p>Ergo, per reductionem ad absurdum habetur<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{1a= a, 1b= b\\}\\vdash 1 = 1^\\prime.<\/span><\/span><\/p>\n<p>Id est, si duo neutra multiplicativa adsint, tunc idem sunt, atque ideo est unicum.<\/p>\n<h4>Valet Lex Simplificationis ad Additionem<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=461s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Hoc est quod facimus cum<\/span><\/strong><\/a> terminos in aequatione removemus.<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = a+c \\longleftrightarrow a = c<\/span><\/span><\/p>\n<p>Non difficile est hanc rem demonstrare; satis est sequentem rationem sequi:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a+b = a+c &amp; \\text{; Praemissa} \\\\\n\n(2) &amp; a+b-a = a+c-a &amp; \\text{; Ex $(1)$, addito $-a$ utrique lateri} \\\\\n\n(3) &amp; (a-a)+b = (a-a)+c &amp; \\text{; Ex $(2)$, commutando et associando} \\\\\n\n(4) &amp; 0+b = 0+c &amp; \\text{; Ex $(3)$ et Inverso Additivo} \\\\\n\n(5) &amp; b = c &amp; \\text{; Ex $(4)$ et Neutro Additivo} \\\\\n\n\\end{array}<\/span>\n<p>Quoniam haec ratio in utramque partem applicari potest eisdem gradibus, habetur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b=a+c \\dashv \\vdash b=c<\/span><\/span><\/p>\n<p>Quod aequivalet dicere<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash a+b=a+c \\longleftrightarrow b=c<\/span><\/span><\/p>\n<h4>Neutrale Additivum Est Simul Absorbens Multiplicativum<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=632s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Hoc simpliciter significat<\/span><\/strong><\/a> quod, pro omni <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> in dominio integritatis, valebit<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\cdot 0 = 0<\/span><\/span><\/p>\n<p>Hoc quoque facile demonstratur, satis est sequentem rationem sequi:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+0) &amp; \\text{; Leges distributivae}\\\\\n\n(2) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+a-a) &amp; \\text{; Ex $(1)$ et Inverso Additivo}\\\\\n\n(3) &amp; a\\cdot a + a\\cdot 0 = a\\cdot a + a\\cdot a - a\\cdot a &amp; \\text{; Ex $(2)$ et Distributivitate}\\\\\n\n(4) &amp;  a\\cdot 0 =  a\\cdot a - a\\cdot a &amp; \\text{; Ex $(3)$ et Simplificatio Additiva}\\\\\n\n(5) &amp;  a\\cdot 0 =  0 &amp; \\text{; Ex $(4)$ et Inverso Additivo}\\\\\n\n\\end{array}<\/span>\n<h4><strong>L\u0113x Sign\u014drum:<\/strong><\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=736s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Productum quantitatum eiusdem signi<\/span><\/strong><\/a> semper est positivum; productum quantitatum contrariorum signorum semper est negativum. Huius proprietatis demonstratio quoque est facilis:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot b = a\\cdot b + 0 &amp; \\text{; Neutrale Additivum}\\\\\n\n(2) &amp; a\\cdot b = a\\cdot b + (a)\\cdot(-b) - (a)\\cdot(-b) &amp; \\text{; Ex $(1)$ et Inverso Additivo}\\\\\n\n(3) &amp; a\\cdot b = a\\cdot (b -b) - (a)\\cdot(-b) &amp; \\text{; Ex $(2)$ et Inverso Additivo}\\\\\n\n(4) &amp; a\\cdot b = a\\cdot 0 + (-a)\\cdot(-b) &amp; \\text{; Ex $(3)$ et Inverso Additivo}\\\\\n\n(5) &amp; a\\cdot b = (-a)\\cdot(-b) &amp; \\text{; Ex $(4)$ et Absorbente Multiplicativo}\\\\\n\n\\end{array}<\/span>\n<p>Igitur: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ab = (-a)(-b)<\/span><\/span><\/p>\n<p>Ad signa contraria, ratio similis est:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot(-b) = a \\cdot (-b) + 0 &amp; \\text{; Neutrale Additivum} \\\\\n\n(2) &amp; a\\cdot(-b) = a \\cdot (-b) + a \\cdot b  -  a \\cdot b  &amp; \\text{; Ex $(1)$ et Inverso Additivo} \\\\\n\n(3) &amp; a\\cdot(-b) = a \\cdot (b-b)  -  a \\cdot b  &amp; \\text{; Ex $(2)$ et Distributivitate} \\\\\n\n(4) &amp; a\\cdot(-b) = a \\cdot 0  -  a \\cdot b  &amp; \\text{; Ex $(3)$ et Inverso Additivo} \\\\\n\n(5) &amp; a\\cdot(-b) = - a \\cdot b  &amp; \\text{; Ex $(4)$ et Absorbente Multiplicativo} \\\\\n\n\\end{array}<\/span>\n<p>Igitur: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b) = -a(b)<\/span><\/span><\/p>\n<h4>Si Productum Duorum Numerorum Est Nulla, Tum Alterutrum Est Nulla<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Alia quoque proprietas quae<\/span><\/strong><\/a> saepe adhibetur est haec:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=0 \\leftrightarrow (a=0 \\vee b=0)<\/span><\/span><\/p>\n<p>Demonstratio eius quoque est facilis:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\{a=0\\} \\models a\\cdot b = 0  &amp; \\textbf{; Absorbens Multiplicativum} \\\\\n\n(2) &amp; \\models a=0 \\rightarrow a\\cdot b = 0  &amp;\\text{; TD$(1)$} \\\\\n\n(3) &amp; \\models \\neg (a\\cdot b = 0 ) \\rightarrow \\neg(a=0) &amp;\\text{; CPI$(2)$} \\\\\n\n(4) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models   \\neg(a=0) &amp;\\text{; RTD$(3)$} \\\\\n\n(5) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models   \\neg(b=0) &amp;\\text{; Similiter $(4)$} \\\\\n\n(6) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models  \\neg(a=0) \\wedge \\neg(b=0) &amp;\\text{; $\\wedge$-int$(4,5)$} \\\\\n\n(7) &amp; \\models (\\neg (a\\cdot b = 0 )) \\rightarrow \\neg(a=0) \\wedge \\neg(b=0)  &amp;\\text{; TD$(6)$} \\\\\n\n(8) &amp; \\models \\neg(\\neg(a=0) \\wedge \\neg(b=0) ) \\rightarrow   (a\\cdot b = 0 ) &amp;\\text{; CPI$(7)$} \\\\\n\n(9) &amp; \\models (a=0 \\vee b=0) \\rightarrow   (a\\cdot b = 0 ) &amp;\\text{; DM$(8)$} \\\\\n\n(10)&amp; \\{a\\neq 0 , a\\cdot b=0\\} \\models b=0 &amp; \\textbf{; Absorbens Multiplicativum} \\\\\n\n(11)&amp; \\{a\\cdot b=0\\} \\models a\\neq 0 \\rightarrow  b=0 &amp; \\text{; TD$(10)$} \\\\\n\n(12)&amp; \\{a\\cdot b=0\\} \\models \\neg(a\\neq 0) \\vee  b=0 &amp; \\text{; $\\rightarrow$-Def$(11)$} \\\\\n\n(13)&amp; \\{a\\cdot b=0\\} \\models a=0 \\vee  b=0 &amp; \\text{; DN$(12)$} \\\\\n\n(14)&amp; \\models (a\\cdot b=0) \\rightarrow (a=0 \\vee  b=0) &amp; \\text{; TD$(13)$} \\\\\n\n(15)&amp; \\models (a\\cdot b=0) \\leftrightarrow (a=0 \\vee  b=0) &amp; \\text{; Ex$(9,14)$}\n\n\\end{array}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KZQ3PXeMlKk?si=O_Hek5KFG853Q6qT\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercitia<\/h2>\n<p>Sint <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> quaecumque elementa dominii integritatis <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span>. Demonstrare oportet sequentia proprietates valere:<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(-a)=(-1)a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=306s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-(a+b)=(-a) + (-b)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=827s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b)=-(ab)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=1213s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-(-a)=a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=1628s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(b-c) = ab - ac<\/span><\/span> <strong>[PROPOSITUM]<\/strong><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)+(b-c) = a-c<\/span><\/span> <strong>[PROPOSITUM]<\/strong><\/li>\n<li>Pro omnibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in D<\/span><\/span> existit unicum <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> tale ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\cdot 1 = a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=2029s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xx = x \\leftrightarrow (x=1 \\vee x=0)<\/span><\/span> <strong>[PROPOSITUM]<\/strong><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Dom\u00ednia Integrit\u00e1tis et Numeri Integri Summarium: Hac in lectione notio Dominii Integritatis introducitur, eiusque momentum in studio algebrae generalis explicatur, et nonnullae proprietates eius maximi momenti per probationes formales demonstrantur. Metas Discendi: Hac lectione peracta, discipulus poterit: Intellegere finem studii algebrae generalis. Intellegere notionem dominii integritatis. Explicare elementa fundamentalia communia inter dom\u00ednia integrit\u00e1tis et n\u00fameros [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32317,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":5,"footnotes":""},"categories":[1310,1298],"tags":[],"class_list":["post-33949","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-generalis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Dom\u00ednia Integrit\u00e1tis et Numeri Integri - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explora Dom\u00ednia Integrit\u00e1tis in Algebra, e\u0101rum propriet\u0101t\u0113s et nexum cum Numer\u012bs Integr\u012bs per demonstration\u0113s et exercitation\u0113s.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/dominia-integritatis-et-numeri-integri\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dom\u00ednia Integrit\u00e1tis et Numeri Integri\" \/>\n<meta property=\"og:description\" content=\"Explora Dom\u00ednia Integrit\u00e1tis in Algebra, e\u0101rum propriet\u0101t\u0113s et nexum cum Numer\u012bs Integr\u012bs per demonstration\u0113s et exercitation\u0113s.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/dominia-integritatis-et-numeri-integri\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-05-09T13:00:27+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-08-01T04:54:18+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/03\/algebrageneral-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Dom\u00ednia Integrit\u00e1tis et Numeri Integri\" \/>\n<meta name=\"twitter:description\" content=\"Explora Dom\u00ednia Integrit\u00e1tis in Algebra, e\u0101rum propriet\u0101t\u0113s et nexum cum Numer\u012bs Integr\u012bs per demonstration\u0113s et exercitation\u0113s.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/03\/algebrageneral.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/dominia-integritatis-et-numeri-integri\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/dominia-integritatis-et-numeri-integri\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Dom\u00ednia Integrit\u00e1tis et Numeri Integri\",\"datePublished\":\"2021-05-09T13:00:27+00:00\",\"dateModified\":\"2025-08-01T04:54:18+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/dominia-integritatis-et-numeri-integri\\\/\"},\"wordCount\":1721,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/dominia-integritatis-et-numeri-integri\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/03\\\/algebrageneral.jpg\",\"articleSection\":[\"Algebra Generalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/dominia-integritatis-et-numeri-integri\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/dominia-integritatis-et-numeri-integri\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/dominia-integritatis-et-numeri-integri\\\/\",\"name\":\"Dom\u00ednia Integrit\u00e1tis et Numeri Integri - 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