abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, 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 domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...

s and

simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a sim ...

s. Although this article discusses the above definition, prime ring may also refer to the minimal non-zero

subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...

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 An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

, and for a characteristic ''p'' field (with ''p'' a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

) the prime ring is the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

of order ''p'' (cf.

Prime field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is wid ...

).Page 90 of

Equivalent definitions

A ring ''R'' is prime

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

the zero ideal 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'' = implies ''A'' = or ''B'' = . *For any two ''right'' ideals ''A'' and ''B'' of ''R'', ''AB'' = implies ''A'' = or ''B'' = . *For any two ''left'' ideals ''A'' and ''B'' of ''R'', ''AB'' = implies ''A'' = or ''B'' = . 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

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 × 2 integer

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

is a prime ring.

Properties

* A

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

is a prime ring if and only if it is an

. * A nonzero ring is prime if and only if the

monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...

of its ideals lacks

zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...

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

Notes

References

* {{DEFAULTSORT:Prime Ring Ring theory