Ideal

A subset I of a ring R that is an additive subgroup and is closed under multiplication by arbitrary elements of R (for all r in R and x in I, rx and xr lie in I); in commutative rings this means r x ∈ I for every r ∈ R and x ∈ I.

A subset I of a ring R that is an additive subgroup of R and is closed under multiplication by arbitrary elements of R (r·x and x·r lie in I for r in R and x in I). Ideals encode divisibility, congruence, and factorization structure of the ambient ring and are the kernels of ring homomorphisms.

A subset I of a ring R that is closed under addition and under multiplication by arbitrary elements of R (r·i and i·r lie in I); in commutative rings this means r·i ∈ I for all r ∈ R, i ∈ I.