Definition A ring consists of a set R on which are defined operations of addition and multiplication that satisfy the following properties:

• the ring is an Abelian group with respect to the operation of addition;

• the operation of multiplication on the ring is associative, and thus x(yz) = (xy)z for all elements x, y and z of the ring.

• the operations of addition and multiplication satisfy the Distributive Law, and thus x(y + z) = xy + xz and (x + y)z = xz + yz for all elements x, y and z of the ring.

Lemma 1.1 Let R be a ring. Then x0 = 0 and 0x = 0 for all elements x of R.

Proof The zero element 0 of R satisfies 0 + 0 = 0. Using the Distributive Law, we deduce that x0 + x0 = x(0 + 0) = x0 and 0x + 0x = (0 + 0)x = 0x.

Thus if we add −(x0) to both sides of the identity x0 + x0 = x0 we see that x0 = 0. Similarly if we add −(0x) to both sides of the identity 0x + 0x = 0x we see that 0x = 0.

Lemma 1.2 Let R be a ring. Then (−x)y = −(xy) and x(−y) = −(xy) for all elements x and y of R.

Proof It follows from the Distributive Law that xy+(−x)y = (x+(−x))y = 0y = 0 and xy + x(−y) = x(y + (−y)) = x0 = 0. Therefore (−x)y = −(xy)  and x(−y) = −(xy).

A subset S of a ring R is said to be a subring of R if 0 ∈ S, a + b ∈ S,  −a ∈ S and ab ∈ S for all a, b ∈ S.

A ring R is said to be commutative if xy = yx for all x, y ∈ R. Not every ring is commutative: an example of a non-commutative ring is provided by the ring of n × n matrices with real or complex coefficients when n > 1.

A ring R is said to be unital if it possesses a (necessarily unique) non-zero multiplicative identity element 1 satisfying 1x = x = x1 for all x ∈ R.

Definition A unital commutative ring R is said to be an integral domain if the product of any two non-zero elements of R is itself non-zero.

Definition A field consists of a set on which are defined operations of addition and multiplication that satisfy the following properties:

• the field is an Abelian group with respect to the operation of addition;

• the non-zero elements of the field constitute an Abelian group with respect to the operation of multiplication;

• the operations of addition and multiplication satisfy the Distributive Law, and thus x(y + z) = xy + xz and (x + y)z = xz + yz for all elements x, y and z of the field.

An examination of the relevant definitions shows that a unital commutative ring R is a field if and only if, given any non-zero element x of R, there exists an element x⁻¹ of R such that xx⁻¹ = 1. Moreover a ring R is a field if and only if the set of non-zero elements of R is an Abelian group with respect to the operation of multiplication.

Lemma 1.3 A field is an integral domain.

Proof A field is a unital commutative ring. Let x and y be non-zero elements of a field K. Then there exist elements x⁻¹ and y⁻¹ of K such that xx⁻¹ = 1 and yy⁻¹ = 1. Then xyy⁻¹x⁻¹ = 1. It follows that xy ≠0, since 0(y⁻¹x⁻¹) =0 and 1 ≠0.

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