In ring theory, a branch of abstract algebra, a **prime ring** is a specific type of ring with certain properties that resemble those of prime numbers in number theory. A ring \( R \) is called **prime** if it is not the zero ring (i.e.
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A prime ring is a nonzero ring whose zero ideal is a prime ideal of a noncommutative ring. Equivalently, the product of any two nonzero two-sided ideals is nonzero. This condition is weaker than absence of zero divisors: a full matrix ring over a division ring is prime even when its size is greater than one. For a commutative ring, primeness is exactly being an integral domain.