It is easy to verify that a ring has no zero divisors iff the left and right cancellation laws hold in .

The set of units forms a group under multiplication.

*Proof.* Use that
for
. Use induction.Â â—»

*Proof.* (ii). If
is the least positive integer such that
,
.

(iii). If
,
, then
implies that
or
, a contradiction.Â â—»

*Proof.* Let
and define multiplication in
by
is a ring with identity
and characteristic zero and the map
given by
is a ring monomorphism. If
, use a similar proof with
and multiplication defined by
Then
.Â â—»

If is any ring, the center of is the set . is easily a subring of but may not be an ideal. A left ideal of that is not or is called a proper left ideal. Observe that if has an identity and is an ideal of , iff . A nonzero ideal of is proper iff contains no units of . A division ring has no proper left or right ideals since every nonzero element of is a unit. The ring of matrices over a division ring has proper left and right ideals, but no proper ideals.

The elements of are called generators of the ideal . If then the ideal is denoted and said to be finitely generated. An ideal generated by a single element is called a principal ideal. A principal ideal ring is a ring in which every ideal is principal. A principal ideal domain is an integral domain and a principal ideal ring.