{ localUrl: '../page/ideal_equals_kernel_of_ring_homomorphism.html', arbitalUrl: 'https://arbital.com/p/ideal_equals_kernel_of_ring_homomorphism', rawJsonUrl: '../raw/5r9.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'ideal_equals_kernel_of_ring_homomorphism', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Ideals are the same thing as kernels of ring homomorphisms', clickbait: '', textLength: '1269', alias: 'ideal_equals_kernel_of_ring_homomorphism', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-08-03 18:30:24', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-08-03 18:30:24', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '1', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '39', text: '[summary: In [3gq ring theory], the notion of "[ideal_ring_theory ideal]" corresponds precisely with the notion of "[5r6 kernel] of [-ring_homomorphism]".]\n\nIn [3gq ring theory], the notion of "[ideal_ring_theory ideal]" corresponds precisely with the notion of "[5r6 kernel] of [-ring_homomorphism]".\n\nThis result is analogous to the fact from [3gd group theory] that [4h6 normal subgroups] are the same thing as [49y kernels of group homomorphisms] ([4h7 proof]).\n\n# Proof\n\n## Kernels are ideals\n\nLet $f: R \\to S$ be a ring homomorphism between rings $R$ and $S$.\nWe claim that the kernel $K$ of $f$ is an ideal.\n\nIndeed, it is clearly a [-576] of the ring $R$ when viewed as just an additive group %%note:That is, after removing the multiplicative structure from the ring.%% because $f$ is a *group* homomorphism between the underlying additive groups, and kernels of *group* homomorphisms are subgroups (indeed, *normal* subgroups). ([4h7 Proof.])\n\nWe just need to show, then, that $K$ is closed under multiplication by elements of the ring $R$.\nBut this is easy: if $k \\in K$ and $r \\in R$, then $f(kr) = f(k)f(r) = 0 \\times r = 0$, so $kr$ is in $K$ if $k$ is.\n\n## Ideals are kernels\n\n[todo: refer to the quotient group, and therefore introduce the quotient ring]', 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: '18258', pageId: 'ideal_equals_kernel_of_ring_homomorphism', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-08-03 18:30:26', auxPageId: 'algebraic_ring', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18256', pageId: 'ideal_equals_kernel_of_ring_homomorphism', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-08-03 18:30:24', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }