Abstract Algebra – 1.2 Homomorphisms

Homomorphisms allow us to compare rings.

Definition 1.2.1 If $A$ and $R$ are (not necessarily commutative) rings, a (ring) homomorphism is a function $\varphi :A\to R$ such that

  • $\varphi(1)=1$,
  • $\varphi (a+a')=\varphi(a)+\varphi(a')$,
  • $\varphi(aa')=\varphi(a)\varphi(a')$.

A ring homomorphism that is also a bijection is called an isomorphism. Rings $A$ and $R$ are called isomorphic, denoted by
$$A\cong R,
if there is an isomorphism $\varphi:A\to R$.

Theorem 1.2.1 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
$$\Phi :R[x_1,\dots,x_n]\to S
with $\Phi(x_i)=s_i$ for all $i$ and $\Phi (r)=\varphi(r)$ for all $r\in R$.

By using mathematical induction, it's easy to prove.

Definition 1.2.2 If $R$ is a commutative ring and $a\in R$, then evaluation at $a$ is the function $e_a:R[x]\to R$, defined by $e_a(f(x))=f^\flat(a)$; that is, $e_a(\sum_{i}r_ix^i)=\sum_i r_ia^i$.

Corollary 1.2.1 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
$$\varphi_*:r_0+r_1x+r_2x^2+\cdots\mapsto \varphi(r_0)+\varphi(r_1)x+\varphi(r_2)x^2+\cdots.
Moreover, $\varphi_*$ is a isomorphism if $\varphi$ is.

Proposition 1.2.1 Let $\varphi:A\to R$ be a homomorphism.

  • $\varphi(a^n)=\varphi(a)^n,\forall a\in A,n\in \mathbb{N}$.
  • If $a\in A$ is a unit, then $\varphi(a)$ is a unit and $\varphi(a^{-1})=\varphi(a)^{-1}$, and so $\varphi(U(A))\subset U(R)$. Moreover, if $\varphi$ is an isomorphism, then $U(A)\cong U(R)$.

Definition 1.2.3 If $\varphi:A\to R$ is a homomorphism, then its kernel is
$$\ker \varphi = \{a\in A;\varphi(a)=0\}
and its image is
$$\operatorname{im}\varphi = \{\varphi(a);a\in A\}.
Definition 1.2.4 An ideal in a commutative ring $R$ is a subset $I$ of $R$ such that

  • $0\in I$
  • if $a,b\in I$, then $a+b\in I$,
  • if $a\in I$ and $r\in R$, then $ra\in I$.

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.

Proposition 1.2.2 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.

Proposition 1.2.3 A homomorphism $\varphi:A\to R$ is an injection if and only if $\ker \varphi=(0)$.

Definition 1.2.5 If $b_1,b_2,\dots,b_n$ lie in $R$, then the set of all linear combinations
$$I=\{r_1b_1+r_2b_2+\dots+r_nb_n;r_i\in R,\text{for all }i\}
is an ideal in $R$. We write $I=(b_1,b_2,\dots,b_n)$ in this case, and we call $I$ the ideal generated by $b_1,b_2,\dots,b_n$. In particular, if $n=1$, then
$$I=(b)=\{rb;r\in R\}
is an ideal in $R$. The ideal $(b)$ (often denoted by $Rb$), consisting of all the multiples of $b$, is called principal ideal generated by $b$.

Theorem 1.2.2 Every ideal $I$ in $\mathbb Z$ is a principal ideal; that is, there is $d\in\mathbb Z$ with $I=(d)$.

Proposition 1.2.4 Let $R$ be a commutative ring and let $a,b\in R$. If $a\mid b$ and $b\mid a$, then $(a)=(b)$.

Definition 1.2.6 Elements $a$ and $b$ in a commutative ring $R$ are associates if there exists a unit $u\in R$ with $b=ua$.

Proposition 1.2.5 Let $R$ be a domain and let $a,b\in R$.

  • $a\mid b$ and $b\mid a$ if and only if $a$ and $b$ are associates.
  • The principal ideals $(a)$ and $(b)$ are equal if and only if $a$ and $b$ are associates.

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