Skip to main content

Algebra-II: Companion notes

Abhay Soman

Section 2.1 Definition

We define mappings between rings with unity that preserves the ring structure.

Definition 2.1.1. (Ring homomorphism).

Let \(R\) and \(S\) be rings with unity. We denote the unity of \(R\) (resp., unity of \(S\)) by \(1_R\) (resp., by \(1_S\)). A function \(\varphi\colon R\to S\) is said to be a ring homomorphism if it satisfy following properties.
  1. [Group homomorphism].
    \(\varphi(r_1+r_2)=\varphi(r_1)+\varphi(r_2)\) for every \(r_1,r_2\in R\)
  2. [Unity maps to unity].
    \(\varphi(1_R)=1_S\)
  3. [Preserves multiplication].
    \(\varphi(r_1r_2)=\varphi(r_1)\varphi(r_2)\) for all \(r_1,r_2\in R\)

Observation 2.1.2.

The first condition in Definition 2.1.1 is that a ring homomorphism is a group homomorphism between \((R,+)\) and \((S,+)\text{.}\) The conditions (2) and (3) in Definition 2.1.1 implies that a ring homomorphism is a monoid homomorphism as well.

Convention 2.1.3.

While considering ring homomorphisms, when context is clear, we often write \(\varphi(1)=1\text{.}\)

Definition 2.1.4. (Kernel of a ring homomorphism).

Let \(\varphi\colon R\to S\) be a ring homomorphism. The kernel of \(\varphi\) is the following set.
\begin{equation*} \ker(\varphi)=\{r\in R:\varphi(r)=0\} \end{equation*}

Observation 2.1.5.

The kernel of a ring homomorphism \(\varphi\colon R\to S\) is a two-sided ideal of \(R\text{.}\)

Definition 2.1.6. (Image of a ring homomorphism).

Let \(\varphi\colon R\to S\) be a ring homomorphism. The image of \(\varphi\) is the following:
\begin{equation*} \Im(\varphi)=\{\varphi(r):r\in R\} \end{equation*}
The \(\Im(\varphi)\) is a subring of \(S\text{.}\)

Observation 2.1.7.

The image of a ring homomorphism \(\varphi\colon R\to S\) need not be an ideal of \(S\text{.}\) For instance, consider the following ring homomorphism from a field \(F\) to a ring of polynomials over \(F\) in one variable:
\begin{equation*} \varphi\colon F\to F[X]\quad\text{given by}\;\alpha\mapsto\alpha\cdot 1 \quad(\text{the constant polynomial }\alpha) \end{equation*}
The image of this map is the set of all constant polynomials, and hence \(\Im(\varphi)\subsetneq F[X]\text{.}\) Since every nonzero element of the image is invertible it can not be a left or a right or a two-sided ideal. Indeed, suppose that \(\Im(\varphi)\) is a left ideal of \(F[X]\text{.}\) Note that every \(0\neq\alpha\in\Im(\varphi)\) is invertible. If \(\beta\in F[X]\) is the inverse of \(\alpha\in F[X]\) then, \(\beta\alpha=1\in\Im(\varphi)\text{.}\) This will imply that if \(\Im(\varphi)\) is a left ideal then \(\Im(\varphi)=F[X]\text{,}\) a contradiction.

Definition 2.1.8. (Monomorphism).

Let \(\varphi\colon R\to S\) be a ring homomorphism. We call \(\varphi\) a monomorphism if \(\ker(\varphi)=\{0\}\text{.}\)

Observation 2.1.9.

Let \(\varphi\colon R\to S\) be a map. The following are equivalent.
  1. \(\varphi\) is an monomorphism
  2. \(\varphi\) is a ring homomorphism and it is injective as a set-theoretic map

Definition 2.1.10. (Epimorphism).

A ring homomorphism \(\varphi\colon R\to S\) is said to be a epimorphism if \(\Im(\varphi)=S\text{.}\)

Definition 2.1.11. (Isomorphism).

Let \(R\) and \(S\) be rings. The ring homomorphism \(\varphi\colon R\to S\) is said to be an isomorphism if there exists a ring homomorphism \(\psi\colon S\to R\) such that \(\psi\circ\phi=\unit_{R}\) and \(\phi\circ\psi=\unit_S\text{,}\) where \(\unit_R\) (resp., \(\unit_S\)) is the identity homomorphism.

Convention 2.1.12.

If two rings \(R\) and \(S\) are isomorphic then we denote it by \(R\simeq S\text{.}\)

Definition 2.1.13. (Automorphism).

A ring isomorphism from a ring onto itself is said to be an automorphism.

Convention 2.1.14.

Let \(R\) and \(S\) be rings. The set of all ring homomorphisms from \(R\) to \(S\) (resp., from \(R\) to \(R\)) is denoted by \(\Hom_{\rm Rings}(R,S)\) (resp., \(\End_{\rm Rings}(R)\)). The set of all ring automorphisms of \(R\) is denoted by \(\Aut_\Rings(R)\text{.}\)

Definition 2.1.15. (Algebra homomorphism).

Let \(F\) be a field. Let \(A\) and \(B\) be \(F\)-algebras. A map \(\varphi\colon A\to B\) is said to be an \(F\)-algebra homomorphism if \(\varphi\) is both an \(F\)-vector space homomorphism and a ring homomorphism.
Hence, if \(a_1,\ldots,a_4\in A\) and \(\alpha,\beta\in F\) then
\begin{equation*} \varphi\left(\alpha\cdot a_1a_2+\beta\cdot a_3a_4\right)=\alpha\cdot\varphi(a_1)\varphi(a_2)+\beta\cdot\varphi(a_3)\varphi(a_4). \end{equation*}
Monomorphisms, epimorphisms, isomorphisms, and automorphisms between \(F\)-algebras are defined in a similar way.

Convention 2.1.16.

Let \(A\) and \(B\) be \(F\)-algebras. Similar to ring case we denote by \(\Hom_{F\text{-Alg}}(A,B)\text{,}\) \(\End_{F\text{-Alg}}(A)\) and \(\Aut_{F\text{-Alg}}(A)\) the sets of all \(F\)-algebra homomorphisms, \(F\)-algebra endomorphisms and \(F\)-algebra automorphisms, respectively.
If \(A\) and \(B\) are isomorphic \(F\)-algebras then we write \(A\simeq B\text{.}\)
<Prev^TopNext>