# Cross product

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crossed product, of a group and a ring An associative ring defined as follows. Suppose one is given a mapping of a group into the isomorphism group of an associative ring with an identity, and a family of invertible elements of , satisfying the conditions  for all and . The family is called a factor system. Then the cross product of and with respect to the factor system and the mapping is the set of all formal finite sums of the form (where the are symbols uniquely assigned to every element ), with binary operations defined by  This ring is denoted by ; the elements form a -basis of it.

If maps onto the identity automorphism of , then is called a twisted or crossed group ring, and if, in addition, for all , then is the group ring of over (see Group algebra).

Let be a field and a monomorphism. Then is a simple ring, being the cross product of the field with its Galois group.

How to Cite This Entry:
Cross product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cross_product&oldid=16831
This article was adapted from an original article by A.A. Bovdi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article