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A non-empty subset I of a ring R that satisfies
x,y ∈ I \implies x−y ∈ I (i.e. (I,+) is a subgroup of (R,+) ;
x ∈ I, r ∈ R \implies rx ∈ I .
Equivalently, there is a ring homomorphism from R which has I as its kernel. Note that this is only itself a ring if it contains 1 , in which case it is in fact equal to R by the second property. A nonstandard notation for " I is an ideal in R " is I \triangleleft R . An ideal M is maximal in R if item M ≠ R (i.e. 1 ∉ M ) item M ⊆ I \triangleleft R \implies I=M or I=R . Equivalently, M is maximal iff R/M is a field. An ideal P is prime in R if
1 ∉ P ;
ab ∈ P \implies a ∈ P or b ∈ P .
Note that a ∈ R is prime (sense 1) iff a ≠ 0 and (a) is a prime ideal. See also see principal ideal domain.