{"id":32322,"date":"2021-05-09T13:00:41","date_gmt":"2021-05-09T13:00:41","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32322"},"modified":"2025-03-03T04:03:47","modified_gmt":"2025-03-03T04:03:47","slug":"integrity-domains-and-integer-numbers","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/integrity-domains-and-integer-numbers\/","title":{"rendered":"Integrity Domains and Integer Numbers"},"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>Integrity Domains and Integer Numbers<\/h1>\n<p style=\"text-align:center;\"><em><strong>Summary:<\/strong><br \/>\nThis class introduces the concept of Integrity Domain, explains its relevance in the study of general algebra, and demonstrates some of its most important properties through formal proofs.<\/em><\/p>\n<p style=\"text-align:center;\"><em><strong>Learning Objectives:<\/strong><\/em><br \/>\nBy the end of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Understand<\/strong> the purpose of studying general algebra. <\/li>\n<li><strong>Understand<\/strong> the concept of an integrity domain. <\/li>\n<li><strong>Explain<\/strong> the basic common aspects between integrity domains and integer numbers. <\/li>\n<li><strong>Demonstrate<\/strong> the basic properties of integrity domains through formal proofs. <\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<strong><u>CONTENT INDEX<\/u><\/strong><br \/>\n<a href=\"#1\">THE PURPOSE OF GENERAL ALGEBRA AND PRIOR KNOWLEDGE<\/a><br \/>\n<a href=\"#2\">FROM INTEGER NUMBERS TO INTEGRITY DOMAINS<\/a><br \/>\n<a href=\"#3\">BASIC COMMON ASPECTS BETWEEN INTEGRITY DOMAINS AND INTEGER NUMBERS<\/a><br \/>\n<a href=\"#4\">PROPERTIES OF INTEGRITY DOMAINS AND INTEGER NUMBERS<\/a><br \/>\n<a href=\"#5\">EXERCISES<\/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>The Purpose of General Algebra and Prior Knowledge<\/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;\">The main objective of general algebra<\/span><\/strong><\/a> is the study of the entire variety of possible mathematical systems. Here we will study several such systems, among the most important of which are natural and integer numbers, and through the latter, we will reach integrity domains.<\/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>From Integer Numbers to Integrity Domains<\/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;\">We will begin our study with integer numbers,<\/span><\/strong><\/a> and the reason for proceeding this way is that they share the most similarities with the majority of the numerical systems we will review in this study.<\/p>\n<p>Instead of trying to define what integer numbers are, we will start by assuming that, whatever they may be, they satisfy certain properties. To do this, a set of axioms is chosen so that it is possible to infer all the properties that we intuitively associate with integers.<\/p>\n<p>All these things are done through the <strong>Peano axioms<\/strong> of the Naturals by introducing the basic operations of arithmetic. Following this axiomatic method and expanding the various operations on natural and integer numbers, new numerical sets emerge, such as rationals, irrationals, reals, complex numbers, quaternions, octonions, and many more.<\/p>\n<p>Then, if we observe integer numbers, we will see that they have properties that will repeat across all other numerical sets, such as the existence of a multiplicative identity, an additive identity, and distributive laws. By referring to these concepts, we can establish a language that allows us to talk about all these sets simultaneously. It is in this context that words like<\/p>\n<ul>\n<li>Integrity Domain<\/li>\n<li>Ring<\/li>\n<li>Group<\/li>\n<li>Vector Space<\/li>\n<\/ul>\n<p>And many more terms of this kind emerge. We will focus our efforts on studying <strong>Integrity Domains<\/strong> first.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Basic Common Aspects of Integrity Domains and Integer Numbers<\/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;\">To explain what an integrity domain is<\/span><\/strong><\/a>, we will rely on the properties that we understand very well from integer numbers. In this context, if <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>, and <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> are integers, then the following <strong>laws<\/strong> hold:<\/p>\n<ol>\n<li><strong>Commutative Laws:<\/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>Associative Laws:<\/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>Distributive Law:<\/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>In addition, there exist certain special elements known as identity elements:<\/p>\n<ul>\n<li><strong>Additive Identity:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = a \\leftrightarrow c=0<\/span><\/span><\/li>\n<li><strong>Multiplicative Identity:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac = a \\leftrightarrow c=1<\/span><\/span><\/li>\n<\/ul>\n<p>The object denoted by <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> is the additive identity, while the one denoted by <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> is the multiplicative identity.<\/p>\n<p>Integers also possess additive inverses. Each integer has an additive inverse that, when added to it, yields the additive identity.<\/p>\n<ul>\n<li><strong>Additive Inverse:<\/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>Additive inverses are recognized by the \u00ab-\u00bb sign that accompanies them.<\/p>\n<p>Finally, there exists a <strong>simplification law<\/strong> expressed through the relation:<\/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>These properties we have reviewed hold for many other sets: real numbers, complex numbers, polynomials, etc. Thus, we call any set that satisfies these properties an <strong>Integrity Domain<\/strong>.<\/p>\n<p><span style=\"color: #800000;\"><strong>DEFINITION:<\/strong><\/span> An Integrity Domain is any set <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> equipped with an addition and multiplication operation such that:<\/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>Additionally, the <strong>associative, commutative,<\/strong> and <strong>distributive laws<\/strong> hold, <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> contains <strong>additive and multiplicative identities<\/strong> (each of these being unique), and finally, the <strong>simplification law<\/strong> holds.<\/p>\n<h4>Example of an Integrity Domain<\/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;\">Consider the set <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>This set, equipped with the usual operations of addition and multiplication, is an integrity domain because it satisfies the laws of commutativity, associativity, and distribution, has additive and multiplicative identities, and finally, an additive inverse.<\/p>\n<ul>\n<li><strong>Additive Identity:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>Multiplicative Identity:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>Additive Inverse:<\/strong> Every element <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b\\sqrt{3}<\/span><\/span> has an additive inverse <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-a-b\\sqrt{3}<\/span><\/span><\/li>\n<\/ul>\n<p>Most importantly, this set A is closed under addition and multiplication, meaning that if we take <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x+y\\in A<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xy\\in A.<\/span><\/span> This is easy to verify: If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_1 + b_1\\sqrt{3}<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_2 + b_2\\sqrt{3}<\/span><\/span> are elements of <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>, then:<\/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>Properties of Integrity Domains and Integer Numbers<\/h3>\n<h4>The Additive Identity of an Integrity Domain is Unique<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=26s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">This can be proven by contradiction:<\/span><\/strong><\/a> Suppose that there exist two additive identities, namely <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0^\\prime<\/span><\/span>. Then, we would have:<\/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{; Premise}\\\\\n\n(2) &amp; a+0 = a &amp; \\text{; Premise: $0$ is the additive identity}\\\\\n\n(3) &amp; b+0^\\prime = b &amp; \\text{; Premise: $0^\\prime$ is the additive identity}\\\\\n\n(4) &amp; 0^\\prime + 0 = 0^\\prime &amp; \\text{; Substituting $a=0^\\prime$ in $(2)$}\\\\\n\n(5) &amp; 0 + 0^\\prime = 0 &amp; \\text{; Substituting $b=0$ in $(3)$}\\\\\n\n(6) &amp; 0 = 0^\\prime  &amp; \\text{; From $(4,5)$ and the commutativity of addition}\\\\\n\n(7) &amp; \\bot &amp;\\text{; From $(1,6)$}\n\n\\end{array}<\/span>\n<p>From this reasoning, we conclude that:<\/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>Thus, by contradiction, we obtain:<\/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>That is, if there are two additive identities, then they must be the same, and therefore, the additive identity is unique.<\/p>\n<h4>The Multiplicative Identity is Also Unique<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=305s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The proof is practically analogous to the previous one.<\/span><\/strong><\/a> If there were two: <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1^\\prime<\/span><\/span>, then we could follow this reasoning:<\/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{; Premise}\\\\\n\n(2) &amp; 1\\cdot a = a &amp; \\text{; Premise: $1$ is the multiplicative identity}\\\\\n\n(3) &amp; 1^\\prime \\cdot b = b &amp; \\text{; Premise: $1^\\prime$ is the multiplicative identity}\\\\\n\n(4) &amp; 1\\cdot 1^\\prime = 1^\\prime &amp; \\text{; Substituting $a=1^\\prime$ in $(2)$}\\\\\n\n(5) &amp; 1^\\prime \\cdot 1 = 1 &amp; \\text{; Substituting $b=1$ in $(3)$}\\\\\n\n(6) &amp; 1 = 1^\\prime  &amp; \\text{; From $(4,5)$ and the commutativity of multiplication}\\\\\n\n(7) &amp; \\bot &amp;\\text{; From $(1,6)$}\n\n\\end{array}<\/span>\n<p>Therefore, we reach the conclusion that:<\/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>Thus, by contradiction, we obtain:<\/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>That is, if there are two multiplicative identities, then they must be the same, and therefore, the multiplicative identity is unique.<\/p>\n<h4>The Simplification Law for Addition Holds<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=461s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">This is what we do when<\/span><\/strong><\/a> we eliminate terms in an equality:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = a+c \\longleftrightarrow b = c<\/span><\/span><\/p>\n<p>It is not difficult to prove this situation; we only need to follow the reasoning below:<\/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{; Premise} \\\\\n\n(2) &amp; a+b-a = a+c-a &amp; \\text{; From $(1)$, subtracting $a$ from both sides} \\\\\n\n(3) &amp; (a-a)+b = (a-a)+c &amp; \\text{; From $(2)$, using commutativity and associativity} \\\\\n\n(4) &amp; 0+b = 0+c &amp; \\text{; From $(3)$ and Additive Inverse} \\\\\n\n(5) &amp; b = c &amp; \\text{; From $(4)$ and Additive Identity} \\\\\n\n\\end{array}<\/span>\n<p>Since this reasoning can be applied in both directions using the same steps, we obtain:<\/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>Which is equivalent to saying:<\/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>The Additive Identity is Also a Multiplicative Absorber<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=632s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">This simply means<\/span><\/strong><\/a> that for every <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> in the integrity domain, the following holds:<\/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>This is also easy to prove; we only need to follow the reasoning below:<\/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{; Distributive Laws}\\\\\n\n(2) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+a-a) &amp; \\text{; From $(1)$ and Additive Inverse}\\\\\n\n(3) &amp; a\\cdot a + a\\cdot 0 = a\\cdot a + a\\cdot a - a\\cdot a &amp; \\text{; From $(2)$ and Distributivity}\\\\\n\n(4) &amp;  a\\cdot 0 =  a\\cdot a - a\\cdot a &amp; \\text{; From $(3)$ and Addition Simplification}\\\\\n\n(5) &amp;  a\\cdot 0 =  0 &amp; \\text{; From $(4)$ and Additive Inverse}\\\\\n\n\\end{array}<\/span>\n<h4><strong>Sign Rule:<\/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;\">The product of numbers with the same sign<\/span><\/strong><\/a> is always positive; the product of numbers with opposite signs is always negative. The proof of this property is also simple:<\/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{; Additive Identity}\\\\\n\n(2) &amp; a\\cdot b = a\\cdot b + (a)\\cdot(-b) - (a)\\cdot(-b) &amp; \\text{; From $(1)$ and Additive Inverse}\\\\\n\n(3) &amp; a\\cdot b = a\\cdot (b -b) - (a)\\cdot(-b) &amp; \\text{; From $(2)$ and Additive Inverse}\\\\\n\n(4) &amp; a\\cdot b = a\\cdot 0 + (-a)\\cdot(-b) &amp; \\text{; From $(3)$ and Additive Inverse}\\\\\n\n(5) &amp; a\\cdot b = (-a)\\cdot(-b) &amp; \\text{; From $(4)$ and Multiplicative Absorber}\\\\\n\n\\end{array}<\/span>\n<p>Therefore: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ab = (-a)(-b)<\/span><\/span><\/p>\n<p>For opposite signs, the reasoning is similar:<\/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{; Additive Identity} \\\\\n\n(2) &amp; a\\cdot(-b) = a \\cdot (-b) + a \\cdot b  -  a \\cdot b  &amp; \\text{; From $(1)$ and Additive Inverse} \\\\\n\n(3) &amp; a\\cdot(-b) = a \\cdot (b-b)  -  a \\cdot b  &amp; \\text{; From $(2)$ and Distributivity} \\\\\n\n(4) &amp; a\\cdot(-b) = a \\cdot 0  -  a \\cdot b  &amp; \\text{; From $(3)$ and Additive Inverse} \\\\\n\n(5) &amp; a\\cdot(-b) = - a \\cdot b  &amp; \\text{; From $(4)$ and Multiplicative Absorber} \\\\\n\n\\end{array}<\/span>\n<p>Therefore: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b) = -a(b)<\/span><\/span><\/p>\n<h4>If the Product of Two Numbers is Zero, at Least One of Them is Zero<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Another property that is also<\/span><\/strong><\/a> widely used is the following:<\/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>Its proof is also simple:<\/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{; Multiplicative Absorber} \\\\\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{; Analogous to $(4)$} \\\\\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{; Multiplicative Absorber}\\\\\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{; From (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>Exercises<\/h2>\n<p>Let <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>, and <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> be any elements of an integrity domain <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span>. Prove that the following properties hold:<\/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;\">[SOLUTION]<\/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;\">[SOLUTION]<\/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;\">[SOLUTION]<\/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;\">[SOLUTION]<\/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>[PROPOSED]<\/strong><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)+(b-c) = a-c<\/span><\/span> <strong>[PROPOSED]<\/strong><\/li>\n<li>For all <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in D<\/span><\/span>, there exists a unique <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> such that <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;\">[SOLUTION]<\/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>[PROPOSED]<\/strong><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Integrity Domains and Integer Numbers Summary: This class introduces the concept of Integrity Domain, explains its relevance in the study of general algebra, and demonstrates some of its most important properties through formal proofs. Learning Objectives: By the end of this class, the student will be able to: Understand the purpose of studying general algebra. [&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":6,"footnotes":""},"categories":[1031,567],"tags":[],"class_list":["post-32322","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-general-algebra","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Integrity Domains and Integer Numbers - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explore Integrity Domains in algebra, their properties, and their relationship with integer numbers through demonstrations and exercises.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/integrity-domains-and-integer-numbers\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integrity Domains and Integer Numbers\" \/>\n<meta property=\"og:description\" content=\"Explore Integrity Domains in algebra, their properties, and their relationship with integer numbers through demonstrations and exercises.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/integrity-domains-and-integer-numbers\/\" \/>\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=\"2021-05-09T13:00:41+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-03-03T04:03:47+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=\"Integrity Domains and Integer Numbers\" \/>\n<meta name=\"twitter:description\" content=\"Explore Integrity Domains in algebra, their properties, and their relationship with integer numbers through demonstrations and exercises.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/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=\"9 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/integrity-domains-and-integer-numbers\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/integrity-domains-and-integer-numbers\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Integrity Domains and Integer Numbers\",\"datePublished\":\"2021-05-09T13:00:41+00:00\",\"dateModified\":\"2025-03-03T04:03:47+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/integrity-domains-and-integer-numbers\\\/\"},\"wordCount\":1921,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/integrity-domains-and-integer-numbers\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/03\\\/algebrageneral.jpg\",\"articleSection\":[\"General Algebra\",\"Mathematics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/integrity-domains-and-integer-numbers\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/integrity-domains-and-integer-numbers\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/integrity-domains-and-integer-numbers\\\/\",\"name\":\"Integrity Domains and Integer Numbers - 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