|
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of , i.e. an element such that :, where is the multiplicative identity. The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the zero ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring. The term ''unit'' is also used to refer to the identity element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also e.g. '''unit' matrix''. For this reason, some authors call "unity" or "identity", and say that is a "ring with unity" or a "ring with identity" rather than a "ring with a unit". The multiplicative identity and its opposite are always units. Hence, pairs of additive inverse elements〔In a ring, the additive inverse of a non-zero element can equal to the element itself.〕 and are always associated. ==Group of units== (詳細はgroup under multiplication, the group of units of . Other common notations for are , , and (from the German term ''Einheit''). In a commutative unital ring , the group of units acts on via multiplication. The orbits of this action are called sets of '; in other words, there is an equivalence relation ∼ on called ''associatedness'' such that : means that there is a unit with . One can check that is a functor from the category of rings to the category of groups: every ring homomorphism induces a group homomorphism , since maps units to units. This functor has a left adjoint which is the integral group ring construction. In an integral domain the cardinality of an equivalence class of associates is the same as that of . A ring is a division ring if and only if . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Unit (ring theory)」の詳細全文を読む スポンサード リンク
|