- Prime ring
In

abstract algebra , a non-trivial ring "R" is a**prime ring**if for any two elements "a" and "b" of "R", if "arb = 0" for all "r" in "R", then either "a = 0" or "b = 0".**Prime ring**can also refer to thesubring of afield determined by its characteristic. For a characteristic 0 field, the prime ring is theinteger s, for a characteristic "p" field (with "p" aprime number ) the prime ring is thefinite field of order "p" (cf.prime field ).cite book |last=Lang |first=Serge |authorlink=Serge Lang |title=Algebra |edition=Third Edition|year=1997 |origyear=1965 |publisher=Addison-Wesley Publishing Company |location=USA |isbn=0-201-55540-9 |pages=p. 90]Prime rings, under the first definition, can be regarded as a simultaneous generalization of both

integral domain s andmatrix ring s over fields.**Examples*** Any domain is a prime ring.

* Anysimple ring is a prime ring, and more generally: every left or rightprimitive ring is a prime ring.

* Any matrix ring over an integral domain is a prime ring. In particular, the ring of 2-by-2 integer matrices is a prime ring.**Properties*** A

commutative ring is a prime ring if and only if it is anintegral domain .

* A ring is prime if and only if its zero ideal is aprime ideal .

* A non-trivial ring is prime if and only if themonoid of its ideals lackszero divisor s.

* The ring of matrices over a prime ring is again a prime ring.**References**

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