Kalpana Kalpana (Editor)

Prime ring

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In abstract algebra, a nonzero ring R is a prime ring if for any two elements a and b of R, arb = 0 for all r in R implies that either a = 0 or b = 0. This definition can be regarded as a simultaneous generalization of both integral domains and simple rings.

Contents

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).

Equivalent definitions

A ring R is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense.

This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for R to be a prime ring:

  • For any two ideals A and B of R, AB={0} implies A={0} or B={0}.
  • For any two right ideals A and B of R, AB={0} implies A={0} or B={0}.
  • For any two left ideals A and B of R, AB={0} implies A={0} or B={0}.

    • Using these conditions it can be checked that the following are equivalent to R being a prime ring:

  • All nonzero right ideals are faithful as right R modules.
  • All nonzero left ideals are faithful as left R modules.

    • 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 nonzero 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

    Prime ring Wikipedia
    (Text) CC BY-SA