nijednostka

Nijednostka is a term used in some Slavic-language mathematical texts to denote a non-unit element of a ring with identity. A unit is an element that has a multiplicative inverse; therefore a nijednostka is an element without such an inverse. In English-language literature the standard term is non-unit. The concept is fundamental in ring theory and in the study of how elements interact under multiplication.

Examples help illustrate the idea. In the ring of integers Z, the only units are 1 and -1, so every other integer is a nijednostka. In the polynomial ring k[x] over a field k, the units are exactly the nonzero constants; polynomials of positive degree are nijednostka. In a local ring, all non-units form the unique maximal ideal, highlighting a structural distinction between units and nijednostki. In a finite ring, an element is a unit if and only if it is not contained in any maximal ideal, so non-units relate closely to the ring’s maximal ideals.

Relations and properties. The set of nijednostki is not generally an ideal, but in a local ring it coincides with the ring’s maximal ideal. The product of units is a unit, and inverses distribute over multiplication, while non-units do not possess inverses. The Jacobson radical of a ring comprises elements that annihilate all simple modules and often includes many nijednostki, especially in non-field rings.

See also: units, ring theory, maximal ideal, local ring, Jacobson radical, ideals. The term nijednostka reflects linguistic usage in Slavic mathematical literature, while in English texts the concept is typically called non-unit.