Definition:Ring (Abstract Algebra)

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This page is about rings in the context of abstract algebra. For other uses, see Definition:Ring.

Definition

A ring $\left({R, *, \circ}\right)$ is a semiring in which $\left({R, *}\right)$ forms an abelian group.


That is, in addition to $\left({R, *}\right)$ being closed, associative and commutative under $*$, it also has an identity, and each element has an inverse.


Ring Axioms

A ring is an algebraic structure $\left({R, *, \circ}\right)$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((A0)\)   $:$   Closure under addition      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b \in R \)             
\((A1)\)   $:$   Associativity of addition      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right) \)             
\((A2)\)   $:$   Commutativity of addition      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b = b * a \)             
\((A3)\)   $:$   Identity element for addition: the zero      \(\displaystyle \exists 0_R \in R: \forall a \in R:\) \(\displaystyle a * 0_R = a = 0_R * a \)             
\((A4)\)   $:$   Inverse elements for addition: negative elements      \(\displaystyle \forall a \in R: \exists a' \in R:\) \(\displaystyle a * a' = 0_R = a' * a \)             
\((M0)\)   $:$   Closure under product      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a \circ b \in R \)             
\((M1)\)   $:$   Associativity of product      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right) \)             
\((D)\)   $:$   Product is distributive over addition      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right) \)             
\(\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right) \)             

These criteria are called the ring axioms.


Note that a ring is still a semiring (in fact, an additive semiring), so all properties of these structures also apply to a ring.


Addition

The distributand $*$ of a ring $\left({R, *, \circ}\right)$ is referred to as ring addition, or just addition.


The conventional symbol for this operation is $+$, and thus a general ring is usually denoted $\left({R, +, \circ}\right)$.


Additive Group

The group $\struct {R, +}$ is known as the additive group of $R$.


Ring Product

The distributive operation $\circ$ in $\left({R, *, \circ}\right)$ is known as the (ring) product.


Binding Priority

In order to simplify expressions involving both $+$ and $\circ$, it is the convention that ring product has a higher precedence than ring addition:

$a \circ b + c := \left({a \circ b}\right) + c$


Ring Less Zero

It is convenient to have a symbol for $R \setminus \set 0$, that is, the set of all elements of the ring without the zero. Thus we usually use:

$R_{\ne 0} = R \setminus \set 0$


Also defined as

Some sources insist on another criterion which a semiring $\left({S, *, \circ}\right)$ must satisfy to be classified as a ring:


\((M2):\quad\) \(\displaystyle \exists 1_R \in R: \forall a \in R:\) \(\) \(\displaystyle a \circ 1_R = a = 1_R \circ a\) $\quad$ Identity element for $\circ$: the unity $\quad$

Such sources refer to what this website calls a ring as a rng (pronounced "rung"): i.e. a "ring" without an "identity".

However, this website specifically defines a ring as one fulfilling axioms $A0 - A4, M0 - M1, D$ only, and instead refers to this more specific structure as a ring with unity.


Other sources define a ring as an algebraic structure $\left({R, *, \circ}\right)$ which, while fulfilling all the other ring axioms, does not insist on $M1$, associativity of ring product.

Such regimes refer to a ring which does fulfil axioms $A0 - A4, M0 - M1, D$ as an associative ring.


Also see

  • If $\left({R^*, \circ}\right)$ is a monoid, then $\left({R, +, \circ}\right)$ is called a ring with unity.
  • If $\left({R^*, \circ}\right)$ is a group, then $\left({R, +, \circ}\right)$ is called a division ring.
  • If $\left({R^*, \circ}\right)$ is an abelian group, then $\left({R, +, \circ}\right)$ is called a field.
  • Results about rings can be found here.


Generalizations


Historical Note

According to Ian Stewart, in his Galois Theory, 3rd ed. of $2004$, the ring axioms were first formulated by Heinrich Martin Weber in $1893$.


Sources