Definition 1

A ring is a set $\mathbf{A}$ with two laws of composition called respectively addition and multiplication, satisfying the following axioms:

(AN I) Under addition $\mathbf{A}$ is a commutative group. (AN II) Multiplication is associative and possesses an identity element. (AN III) Multiplication is distributive with respect to addition. The ring $\mathbf{A}$ is said to be commutative if its multiplication is commutative.

In what follows, $(x, y) \mapsto x + y$ denotes addition and $(x, y) \mapsto xy$ multiplication; $0$ denotes the identity element for addition and $1$ that for multiplication. Finally, $-x$ denotes the negative of $x$ under addition. The axioms of a ring are therefore expressed by the following identities:

Finally the ring $\mathbf{A}$ is commutative if $xy = yx$ for $x, y$ in $\mathbf{A}$.

With addition alone $\mathbf{A}$ is a commutative group called the additive group of $\mathbf{A}$. For all $x \in \mathbf{A}$, we define the left homothety $\gamma_x$ and right homothety $\delta_x$ by $\gamma_x(y) = xy$, $\delta_x(y) = yx$. By formulae (7) and (8), $\gamma_x$ and $\delta_x$ are endomorphisms of the additive group of $\mathbf{A}$ and thus map zero to zero and negative to negative. Therefore

it follows that $(-x)(-y) = -((-x).y) = -(-xy)$, whence Formulae (10) and (11) constitute the sign rule. It follows that and $(-1)(-1) = 1$.

From (11) it follows by induction on $n$ that

When we speak of cancellable elements, invertible elements, permutable elements, central elements, centralizer or centre of a ring $\mathbf{A}$, all these notions will refer to the multiplication on $\mathbf{A}$. If $x, y \in \mathbf{A}$ and $y$ is invertible, the element $xy^{-1}$ of $\mathbf{A}$ is also denoted by $x/y$ when $\mathbf{A}$ is commutative. The set of invertible elements of $\mathbf{A}$ is stable under multiplication. Under the law induced by multiplication it is a group called the multiplicative group of $\mathbf{A}$, sometimes denoted by $\mathbf{A}^*$.

Let $x, y$ be in $\mathbf{A}$. $x$ is said to be a left (resp. right) multiple of $y$ if there exists $y' \in \mathbf{A}$ such that $x = y'y$ (resp. $x = yy'$); it is also said that $y$ is a right (resp.

left) divisor of x. When A is commutative, there is no need to distinguish between "left" and "right".

In conformity with the above terminology, every element $y \in \mathbf{A}$ would be considered as a right and left divisor of 0; but, by an abuse of language, in general the term "right (resp. left) divisor of 0" is reserved for elements $y$ such that there exists $x \neq 0$ in A satisfying the relation $xy = 0$ (resp. $yx = 0$). In other words, the right (resp. left) divisors of zero are the right (resp. left) non-cancellable elements.

Let $x \in \mathbf{A}$. $x$ is called nilpotent if there exists an integer $n > 0$ with $x^n = 0$. The element $1 - x$ is then invertible, with inverse equal to

As A is a commutative group under addition, the element $nx$ for $n \in \mathbf{Z}$ and $x \in \mathbf{A}$ has been defined (§ 2, no. 8). As $\gamma_x$ and $\delta_x$ are endomorphisms of the additive group A, $\gamma_x(ny) = n\gamma_x(y)$ and $\delta_y(nx) = n\delta_y(x)$, whence

In particular, $nx = (n.1)x$.

A set A with addition and multiplication satisfying the axioms of a ring with the exception of that assuring the existence of the identity element under multiplication, is called a pseudo-ring.