#### Glasnik Matematicki, Vol. 37, No.1 (2002), 13-19.

### NOTES ON GALOIS ALGEBRAS

### George Szeto and Lianyong Xue

Department of Mathematics, Bradley University, Peoria, Illinois 61625, USA

*e-mail:* `szeto@hilltop.bradley.edu`

*e-mail:* `lxue@hilltop.bradley.edu`

**Abstract.** Let *B* be a ring with 1, *C* the
center of *B*, and *G* an automorphism group of *B*
of order *n* for some integer *n*. Assume *B* is a
Galois algebra over *R* with Galois group *G*. For a
nonzero idempotent *e*
∈ *R*, it the rank of *Be*
over *Ce* is defined and equal to the order of
*H*|_{Be} where
*H* = {*g*
∈ *G* | *g*(*c*) = *c* for each *c*
∈ *C*}, then
*Be* is a central Galois algebra with Galois group
*H*|_{Be}. This generalizes the F. R.
DeMeyer and T. Kanzaki theorems for Galois algebras. Moreover,
a structure theorem for a Galois algebra is given in terms of the
concept of the rank of projective module.

**2000 Mathematics Subject Classification.**
16S35, 16W20.

**Key words and phrases.** Galois extensions, Galois
algebras, central Galois extensions, separable extensions,
Azumaya algebras, rank of a projective module.

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