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In algebra, a unit or invertible element [a] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that = =, where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u.
One – I (I is the Roman numeral for 1) or rarely A, AN (not normally used in British crosswords), ACE (playing card), UNIT; Operating system – OS; Operation – OP; Order – OM (Order of Merit) Ordinary Seaman – OS, Rating; Oriental – E (East) Other Ranks – OR (military term for non-commissioned ranks) Ounce – OZ (abbreviation ...
The set of units of a ring is a group under ring multiplication; this group is denoted by R × or R* or U(R). For example, if R is the ring of all square matrices of size n over a field, then R × consists of the set of all invertible matrices of size n , and is called the general linear group .
n generate the group of cyclotomic units. If n is a composite number having two or more distinct prime factors, then ζ a n − 1 is a unit. The subgroup of cyclotomic units generated by (ζ a n − 1)/(ζ n − 1) with (a, n) = 1 is not of finite index in general. [2] The cyclotomic units satisfy distribution relations.
A ring R is called unit regular if for every a in R, there is a unit u in R such that a = aua. Every semisimple ring is unit regular, and unit regular rings are directly finite rings. An ordinary von Neumann regular ring need not be directly finite. A ring R is called strongly von Neumann regular if for every a in R, there is some x in R with a ...
The category of rings is a symmetric monoidal category with the tensor product of rings ⊗ Z as the monoidal product and the ring of integers Z as the unit object. It follows from the Eckmann–Hilton theorem, that a monoid in Ring is a commutative ring. The action of a monoid (= commutative ring) R on an object (= ring) A of Ring is an R-algebra.
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In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. [1] It determines the rank of the group of units in the ring O K of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.