## [Ideal](/ideal-2) Definition 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. ## Principle Principle An ideal abstracts the notion of a 'kernel-like' subset stable under ring operations so that one can form quotient rings R/I with well-defined addition and multiplication. ## Demonstration Demonstration In Z, the set nZ of multiples of an integer n is an ideal; in k[x], the set (f) of all multiples of a polynomial f is a principal ideal generated by f. ## Misapplication Misapplication Confusing an ideal with a subring: an ideal need not contain the multiplicative identity 1 and typically is not closed under multiplication of its own elements unless that follows from the ring's multiplication. ## Consequence Consequence Ideals allow formation of quotient rings and support constructions such as primary decomposition, localizations, and the study of homomorphism kernels. ## Reversal Reversal A multiplicative subset or a subring that contains 1 and is closed under multiplication but not under additive absorption differs from an ideal; such subsets do not yield quotients by the same mechanism. ## Boundary Boundary Distinguish two-sided, left, and right ideals in noncommutative rings; in commutative algebra 'ideal' usually means two-sided. Ideals are subsets of rings, not of modules (though ideals are modules over the ring). ## Semantic Tension Semantic Tension Ideal vs submodule: an ideal is a submodule of the ring viewed as a module over itself, but thinking of ideals only as subrings or only as kernels can obscure their absorption property under ring multiplication. ## Synthesis Synthesis An ideal is a subset of a ring closed under addition and absorption by ring elements, serving as the appropriate notion of 'kernel' that permits quotienting and analysis of ring structure.