Prime ring


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 the subring of a field determined by its characteristic. For a characteristic 0 field, the prime ring is the integers, for a characteristic "p" field (with "p" a prime number) the prime ring is the finite 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 domains and matrix rings over fields.

Examples

* Any domain is a prime ring.
* Any simple ring is a prime ring, and more generally: every left or right primitive 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 an integral domain.
* A ring is prime if and only if its zero ideal is a prime ideal.
* A non-trivial ring is prime if and only if the monoid of its ideals lacks zero divisors.
* The ring of matrices over a prime ring is again a prime ring.

References


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