# Prime ring

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.

Although this article discusses the above definition, **prime ring** may also refer to the minimal non-zero subring of a field, which is generated by its identity element 1, and 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).[1]

## 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.

## Notes

- Page 90 of Lang, Serge (1993),
*Algebra*(Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001

## References

- Lam, Tsit-Yuen (2001),
*A First Course in Noncommutative Rings*(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0, MR 1838439