Associative rings and algebras Definition

Ring

Builds on Group

Definition

A ring is a set $R$ with two binary operations and two elements $0, 1 \in R$ that satisfy the axioms below. The operations are the addition $+$ and the multiplication $\cdot$ .

$(R, +)$ is an abelian group with identity element $0$ .
The multiplication is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in R$ .
The element $1$ is a multiplicative identity: $1 \cdot a = a \cdot 1 = a$ for all $a \in R$ .
The multiplication is distributive over the addition: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(b + c) \cdot a = b \cdot a + c \cdot a$ for all $a, b, c \in R$ .

Remarks

Some authors do not require the element $1$ . They call the structure of this definition a ring with identity, or a unital ring. The axioms give $0 \cdot a = a \cdot 0 = 0$ and $(-a) \cdot b = -(a \cdot b)$ for all $a, b \in R$ . The integers $\mathbb{Z}$ , the integers modulo $n$ and the $n \times n$ real matrices are rings. A ring in which $a \cdot b = b \cdot a$ for all $a, b$ is a commutative ring.

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Tag	`0894J980RK`
URI	`m2://0894J980RK`
Kind	definition
Writing	W1 stated
MSC	16B99
Defines	Ring
Uses	Group

Ring

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