Rings

Overview

A ring $⟨R,+,\cdot ⟩$ is a rng with a multiplicative identity $1\in R$ called unity. In other words, $⟨R,\cdot ⟩$ is a monoid.

An element in $R$ is a unit of $R$ if it has a multiplicative inverse in $R$. The multiplicative inverse of an element $a$, if it exists, is denoted $a^{-1}$.

A ring in which multiplication is commutative is a commutative ring.

Subrings

Let $⟨R,+,\cdot ⟩$ be a ring. Then $S$ is a subring of $R$ if:

• $S$ is a subrng of $R$, and
• $1\in R$ is a member of $S$.

Criterion

A nonempty subset $S$ of a ring $R$ is a subring if and only if

• $S$ contains $1\in R$,
• $S$ is closed under subtraction, and
• $S$ is closed under multiplication.

Intersection

Let $\{S_i\mid i\in I\}$ be an indexed set of subrings of $R$. Then their intersection $\bigcap _{i\in I}{S}_i$ is a subring of $R$.

Homomorphisms

Let $R$ and $R^{\prime }$ be rings. A ring homomorphism $\varphi :R\to R^{\prime }$ is a mapping that satisfies the following conditions for all $a,b\in R$:

1. $\varphi (a+b)=\varphi (a){+}^{\prime }\varphi (b)$
2. $\varphi (ab)=\varphi (a)\varphi (b)$
3. $\varphi (1)=1^{\prime }$

Note that $\varphi$ is also a rng homomorphism so the linked structural properties are preserved. Additionally,

• Inverses
  • If $a\in R$ is a unit, then $\varphi (a^{-1})=\varphi (a{)}^{-1}$.
• Subrings
  • If $S\le R$, then $\varphi [[S]]\le R^{\prime }$.
  • If $S^{\prime }\le R^{\prime }$, then ${\varphi }^{-1}[[S]]\le R$.

Direct Products

Let $R_1,R_2,\dots ,R_n$ be rings. Let $a_i,b_i\in R_i$ for $1\le i\le n$ and define

Then the direct product $\prod _{i=1}^n{R}_i$ is a ring under these binary operations.

Characteristic

Let $R$ be a ring. If $n\cdot 1\ne 0$ for all $n\in {\mathbb{Z}}^{+}$, then $R$ has characteristic $0$. If $n\cdot 1=0$ for some $n\in {\mathbb{Z}}^{+}$, then the smallest such integer $n$ is the characteristic of $R$.

For any subring $S$ of $R$, the above immediately implies $\text{char}(R)=\text{char}(S)$.