nLab
group of units

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Contents

Definition

For R a ring, its group of units, denoted R × or GL 1(R), is the group whose elements are the elements of R that are invertible under the product, and whose group operation is the multiplication in R.

Properties

Remark

There is an adjunction

(R[−]⊣(−) ×):Alg R→(−) ×←R[−]Grp(R[-]\dashv (-)^\times) \colon Alg_R \stackrel{\overset{R[-]}{\leftarrow}}{\underset{(-)^\times}{\to}} Grp

between the category of associative algebras over R and that of groups, where R[−] forms the group algebra over R and where (−) × assigns to an R-algebra its group of units.

Revised on October 14, 2012 16:39:05 by Urs Schreiber (89.204.130.129)
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