Irreducible ring

Symbol-free definition
A commutative unital ring is termed irreducible if there do no exist a pair of nonzero ideals whose intersection is the trivial ideal. Equivalently, a ring is irreducible if the zero ideal is an irreducible ideal.

Definition with symbols
A commutative unital ring $$R$$ is termed irreducible if whenever $$I \cap J = 0$$ for ideals $$I$$ and $$J$$ of $$R$$, then either $$I=0$$ or $$J=0$$.

Stronger properties

 * Field
 * Integral domain