nichtideale

Nichtideale is a German term that translates to non-ideals. In mathematical writing, it is used informally to refer to subsets of a ring or algebraic structure that do not satisfy the defining properties of an ideal. An ideal I of a ring R must be closed under addition and under multiplication by arbitrary ring elements: for all a, b in I, a + b is in I, and for all r in R and i in I, ri and ir are in I. A Nichtideal, therefore, is any subset of R that fails at least one of these conditions.

Etymology and usage: The word combines nicht (not) with Ideale (ideals). While not a standardized technical term with a formal definition in most texts, Nichtideale are commonly mentioned to illustrate what is not an ideal, or to contrast with true ideals in discussions of ring structure, quotient constructions, or module theory. In careful writings, authors will specify that the discussion concerns “subsets that are not ideals” rather than naming a separate mathematical class.

Examples: In the ring of integers Z, the set 2Z of even integers is an ideal. The subset {0, 2} of Z is a Nichtideal because it is not closed under addition (2 + 2 = 4 is not in {0, 2}) and does not absorb multiplication by all elements of Z. The natural numbers N viewed inside Z are also not an ideal, since they are not closed under negation.

See also: Ideale, Ringtheorie, Unterringe, Untermodul.