Homomorphisms allow us to compare rings.<\/p>\n
Definition 1.2.1<\/strong> If $A$ and $R$ are (not necessarily commutative) rings, a (ring) homomorphism<\/strong> is a function $\\varphi :A\\to R$ such that<\/p>\n A ring homomorphism that is also a bijection is called an isomorphism<\/strong>. Rings $A$ and $R$ are called isomorphic<\/strong>, denoted by Theorem 1.2.1<\/strong> Let $R$ and $S$ be commutative rings, and let $\\varphi:R\\to S$ be a homomorphism, and let $\\varphi:R\\to S$ be a homomorphism. If $s_1,\\dots,s_n\\in S$, then there exists a unique homomorphism By using mathematical induction, it’s easy to prove.<\/p>\n Definition 1.2.2<\/strong> If $R$ is a commutative ring and $a\\in R$, then evaluation at<\/strong> $a$ is the function $e_a:R[x]\\to R$, defined by $e_a(f(x))=f^\\flat(a)$; that is, $e_a(\\sum_{i}r_ix^i)=\\sum_i r_ia^i$. <\/p>\n Corollary 1.2.1<\/strong> If $R$ and $S$ are commutative rings and $\\varphi: R\\to S$ is a homomorphism, then there is a homomorphism $\\varphi_*:R[x]\\to S[x]$ given by Proposition 1.2.1<\/strong> Let $\\varphi:A\\to R$ be a homomorphism.<\/p>\n Definition 1.2.3<\/strong> If $\\varphi:A\\to R$ is a homomorphism, then its kernel<\/strong> is The ring $R$ itself and $(0)$, the subset consisting $0$ alone, are always ideals in a commutative ring $R$. An ideal $I\\ne R$ is called a proper ideal<\/strong>.<\/p>\n Proposition 1.2.2<\/strong> If $\\varphi:A\\to R$ is a homomorphism, then $\\ker \\varphi$ is an ideal in $A$ and $\\operatorname{im} \\varphi$ is a subring of $R$. Moreover, if $A$ and $R$ are not zero rings, then $\\ker\\varphi$ is a proper ideal.<\/p>\n Proposition 1.2.3<\/strong> A homomorphism $\\varphi:A\\to R$ is an injection if and only if $\\ker \\varphi=(0)$.<\/p>\n Definition 1.2.5<\/strong> If $b_1,b_2,\\dots,b_n$ lie in $R$, then the set of all linear combinations Theorem 1.2.2<\/strong> Every ideal $I$ in $\\mathbb Z$ is a principal ideal; that is, there is $d\\in\\mathbb Z$ with $I=(d)$.<\/p>\n Proposition 1.2.4<\/strong> Let $R$ be a commutative ring and let $a,b\\in R$. If $a\\mid b$ and $b\\mid a$, then $(a)=(b)$.<\/p>\n Definition 1.2.6<\/strong> Elements $a$ and $b$ in a commutative ring $R$ are associates<\/strong> if there exists a unit $u\\in R$ with $b=ua$.<\/p>\n Proposition 1.2.5<\/strong> Let $R$ be a domain and let $a,b\\in R$.<\/p>\n Homomorphisms allow us to compare rings. Definition 1.2 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,6],"tags":[11,8],"class_list":["post-398","post","type-post","status-publish","format-standard","hentry","category-abstract-algebra","category-6","tag-11","tag-8"],"_links":{"self":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/398","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=398"}],"version-history":[{"count":2,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/398\/revisions"}],"predecessor-version":[{"id":400,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/398\/revisions\/400"}],"wp:attachment":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=398"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=398"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=398"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\n$$A\\cong R,
\n$$
\nif there is an isomorphism $\\varphi:A\\to R$.<\/p>\n
\n$$\\Phi :R[x_1,\\dots,x_n]\\to S
\n$$
\nwith $\\Phi(x_i)=s_i$ for all $i$ and $\\Phi (r)=\\varphi(r)$ for all $r\\in R$.<\/p>\n
\n$$\\varphi_*:r_0+r_1x+r_2x^2+\\cdots\\mapsto \\varphi(r_0)+\\varphi(r_1)x+\\varphi(r_2)x^2+\\cdots.
\n$$
\nMoreover, $\\varphi_*$ is a isomorphism if $\\varphi$ is.<\/p>\n\n
\n$$\\ker \\varphi = \\{a\\in A;\\varphi(a)=0\\}
\n$$
\nand its image<\/strong> is
\n$$\\operatorname{im}\\varphi = \\{\\varphi(a);a\\in A\\}.
\n$$
\nDefinition 1.2.4<\/strong> An ideal<\/strong> in a commutative ring $R$ is a subset $I$ of $R$ such that<\/p>\n\n
\n$$I=\\{r_1b_1+r_2b_2+\\dots+r_nb_n;r_i\\in R,\\text{for all }i\\}
\n$$
\nis an ideal in $R$. We write $I=(b_1,b_2,\\dots,b_n)$ in this case, and we call $I$ the ideal generated by<\/strong> $b_1,b_2,\\dots,b_n$. In particular, if $n=1$, then
\n$$I=(b)=\\{rb;r\\in R\\}
\n$$
\nis an ideal in $R$. The ideal $(b)$ (often denoted by $Rb$), consisting of all the multiples of $b$, is called principal ideal<\/strong> generated by $b$.<\/p>\n\n