Basic ring theory

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. ◻

Proof. (ii). If is the least positive integer such that , .
(iii). If , , then implies that or , a contradiction. ◻

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 . ◻

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.