# Cross product

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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.

#### References

 [1] S.K. Sehgal, "Topics in group rings" , M. Dekker (1978) [2] A.A. Bovdi, "Cross products of semi-groups and rings" Sibirsk. Mat. Zh. , 4 (1963) pp. 481–499 (In Russian) [3] A.E. Zalesskii, A.V. Mikhalev, "Group rings" J. Soviet Math. , 4 (1975) pp. 1–74 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 2 (1973) pp. 5–118 [4] D.S. Passman, "The algebraic structure of group rings" , Wiley (1977)