{ localUrl: '../page/kernel_of_ring_homomorphism.html', arbitalUrl: 'https://arbital.com/p/kernel_of_ring_homomorphism', rawJsonUrl: '../raw/5r6.json', likeableId: '3324', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'kernel_of_ring_homomorphism', edit: '4', editSummary: '', prevEdit: '3', currentEdit: '4', wasPublished: 'true', type: 'wiki', title: 'Kernel of ring homomorphism', clickbait: 'The kernel of a ring homomorphism is the collection of things which that homomorphism sends to 0.', textLength: '1537', alias: 'kernel_of_ring_homomorphism', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-08-04 19:38:29', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-08-03 16:52:59', 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: '27', text: '[summary: The kernel of a [-ring_homomorphism] is the collection of elements which the homomorphism sends to $0$.]\n\nGiven a [-ring_homomorphism] $f: R \\to S$ between [3gq rings] $R$ and $S$, we say the **kernel** of $f$ is the collection of elements of $R$ which $f$ sends to the zero element of $S$.\n\nFormally, it is $$\\{ r \\in R \\mid f(r) = 0_S \\}$$\nwhere $0_S$ is the zero element of $S$.\n\n# Examples\n\n- Given the "identity" (or "do nothing") ring homomorphism $\\mathrm{id}: \\mathbb{Z} \\to \\mathbb{Z}$, which sends $n$ to $n$, the kernel is just $\\{ 0 \\}$.\n- Given the ring homomorphism $\\mathbb{Z} \\to \\mathbb{Z}$ taking $n \\mapsto n \\pmod{2}$ (using the usual shorthand for [-5ns]), the kernel is the set of even numbers.\n\n# Properties\n\nKernels of ring homomorphisms are very important because they are precisely [ideal_ring_theory ideals]. ([5r9 Proof.])\nIn a way, "ideal" is to "ring" as "[-576]" is to "[3gd group]", and certainly [subring_ring_theory subrings] are much less interesting than ideals; a lot of ring theory is about the study of ideals.\n\nThe kernel of a ring homomorphism always contains $0$, because a ring homomorphism always sends $0$ to $0$.\nThis is because it may be viewed as a [-47t] acting on the underlying additive group of the ring in question, and [49z the image of the identity is the identity] in a group.\n\nIf the kernel of a ring homomorphism contains $1$, then the ring homomorphism sends everything to $0$.\nIndeed, if $f(1) = 0$, then $f(r) = f(r \\times 1) = f(r) \\times f(1) = f(r) \\times 0 = 0$.', 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: '18369', pageId: 'kernel_of_ring_homomorphism', userId: 'PatrickStevens', edit: '4', type: 'newEdit', createdAt: '2016-08-04 19:38:29', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18238', pageId: 'kernel_of_ring_homomorphism', userId: 'PatrickStevens', edit: '3', type: 'newEdit', createdAt: '2016-08-03 16:54:18', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18237', pageId: 'kernel_of_ring_homomorphism', userId: 'PatrickStevens', edit: '2', type: 'newEdit', createdAt: '2016-08-03 16:53:14', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18236', pageId: 'kernel_of_ring_homomorphism', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-08-03 16:53:01', auxPageId: 'algebraic_ring', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18234', pageId: 'kernel_of_ring_homomorphism', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-08-03 16:52:59', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }