{ localUrl: '../page/integral_domain.html', arbitalUrl: 'https://arbital.com/p/integral_domain', rawJsonUrl: '../raw/5md.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'integral_domain', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Integral domain', clickbait: 'An integral domain is a ring where the only way to express zero as a product is by having zero as one of the terms.', textLength: '2884', alias: 'integral_domain', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-07-28 13:33:14', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-07-28 13:33:14', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '60', text: '[summary: An integral domain is a [3gq ring] in which the only way to make $0$ as a product is to multiply $0$ by something. For instance, in an integral domain like [48l $\\mathbb{Z}$], $2 \\times 3$ is not equal to $0$ because neither $2$ nor $3$ is.]\n\n[summary(Technical): An integral domain is a [3gq ring] in which $ab=0$ implies $a=0$ or $b=0$. (We exclude the ring with one element: that is conventionally not considered an integral domain.)]\n\nIn keeping with [3gq ring theory] as the attempt to isolate each individual property of [48l $\\mathbb{Z}$] and work out how the properties interplay with each other, we define the notion of **integral domain** to capture the fact that if $a \\times b = 0$ then $a=0$ or $b=0$.\nThat is, an integral domain is one which has no "zero divisors": $0$ cannot be nontrivially expressed as a product.\n(For uninteresting reasons, we also exclude the ring with one element, in which $0=1$, from being an integral domain.)\n\n# Examples\n\n- $\\mathbb{Z}$ is an integral domain.\n- Any [481 field] is an integral domain. (The proof is an exercise.)\n\n%%hidden(Show solution):\nSuppose $ab = 0$, but $a \\not = 0$. We wish to show that $b=0$.\n\nSince we are working in a field, $a$ has an inverse $a^{-1}$; multiply both sides by $a^{-1}$ to obtain $a^{-1} a b = 0 \\times a^{-1}$.\nSimplifying, we obtain $b = 0$.\n%%\n\n- When $p$ is a [4mf prime] integer, the ring $\\mathbb{Z}_p$ of integers [modular_arithmetic mod] $p$ is an integral domain.\n- When $n$ is a [composite_number composite] integer, the ring $\\mathbb{Z}_n$ is *not* an integral domain. Indeed, if $n = r \\times s$ with $r, s$ positive integers, then $r s = n = 0$ in $\\mathbb{Z}_n$.\n\n# Properties\n\nThe reason we care about integral domains is because they are precisely the rings in which we may cancel products: if $a \\not = 0$ and $ab = ac$ then $b=c$.\n%%hidden(Proof):\nIndeed, if $ab = ac$ then $ab-ac = 0$ so $a(b-c) = 0$, and hence (in an integral domain) $a=0$ or $b=c$.\n\nMoreover, if we are not in an integral domain, say $r s = 0$ but $r, s \\not = 0$.\nThen $rs = r \\times 0$, but $s \\not = 0$, so we can't cancel the $r$ from both sides.\n%%\n\n## Finite integral domains\n\nIf a ring $R$ is both finite and an integral domain, then it is a [481 field].\nThe proof is an exercise.\n%%hidden(Show solution):\nGiven $r \\in R$, we wish to find a multiplicative inverse.\n\nSince there are only finitely many elements of the ring, consider $S = \\{ ar : a \\in R\\}$.\nThis set is a subset of $R$, because the multiplication of $R$ is [3gy closed].\nMoreover, every element is distinct, because if $ar = br$ then we can cancel the $r$ (because we are in an integral domain), so $a = b$.\n\nSince there are $|R|$-many elements of the subset $S$ (where $| \\cdot |$ refers to the [-4w5]), and since $R$ is finite, $S$ must in fact be $R$ itself.\n\nTherefore in particular $1 \\in S$, so $1 = ar$ for some $a$.\n%%\n', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'PatrickStevens' ], childIds: [], parentIds: [ 'algebraic_ring' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17637', pageId: 'integral_domain', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-07-28 13:33:15', auxPageId: 'algebraic_ring', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17635', pageId: 'integral_domain', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-07-28 13:33:14', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }