X-inner automorphism

-inner automorphisms were introduced by V.K. Kharchenko in [a2] and [a3] to study both prime rings satisfying generalized identities and the Galois theory of semi-prime rings (cf. also Prime ring; Rings and algebras). Since the appropriate definitions are much simpler when the ring is assumed to be prime, this special case is treated first here. Let be a prime ring (with ) and let denote its symmetric Martindale ring of quotients. Then any automorphism of extends uniquely to an automorphism of , and one says that is -inner if is inner on (cf. also Inner automorphism). It is easy to see that , the set of all -inner automorphisms of , is a normal subgroup of .

-inner automorphisms control the generalized linear identities of , namely those linear identities which involve automorphisms. For example, it is shown in [a2] that if and if with for all , then there exists a unit with , and for all . In particular, is the inner automorphism of induced by and consequently is -inner. Of course, is determined by up to multiplication by a non-zero element of the extended centroid .

Now, let be a group of automorphisms of and let , so that . If denotes the linear span of all units in such that conjugation by belongs to , then is a -subalgebra of , called the algebra of the group (cf. also Group algebra). One says that is an -group (Maschke group) if and if is a finite-dimensional semi-simple -algebra. Furthermore, is an -group (Noether group) if is an -group and if conjugation by every unit of induces an automorphism of contained in . The Galois theory, as developed in [a3] and [a14], involves the action of -groups and -groups on prime rings.

Note that if is an -group and if is a unit of , then conjugation by need not stabilize . Thus, it is not always possible to embed an -group into an -group. One can avoid this difficulty by extending the definition of "automorphism of R" to include those (real) automorphisms of such that for some .

-inner automorphisms also appear prominently in the study of cross products. For example, it is proved in [a1] that if is a cross product over the prime ring , then embeds naturally into and that is a twisted group algebra with . Furthermore, it is shown in [a9] that every non-zero ideal of meets non-trivially, and in [a11] that is prime (or semi-prime) if and only if is -prime (or -semi-prime) for all finite normal subgroups of contained in . The above-mentioned structure of is also used in [a5] and [a6] to precisely describe the prime ideals in cross products of finite and of polycyclic-by-finite groups.

There are numerous computations of in the literature. To start with, it is shown in [a4] that if is a non-commutative free algebra, then . More general free products are studied in [a7] and [a8]. Next, [a10] effectively handles graded domains like enveloping algebras of Lie algebras, and [a15] considers arbitrary enveloping algebra smash products. Finally, [a12] and [a13] study certain group algebras and show that for any group there exists a domain with .

Now suppose that is a semi-prime ring and again let denote its symmetric Martindale ring of quotients. If is an arbitrary automorphism of , write . Then, following [a2], one says that is -inner if . Of course, is -outer when . Note that, in the case of semi-prime rings, need not be a subgroup of . Nevertheless, a good deal of structure still exists. For example, [a3] proves the key fact that is always a cyclic -module.

References

[a1]	J.W. Fisher, S. Montgomery, "Semiprime skew group rings" J. Algebra , 52 (1978) pp. 241–247
[a2]	V.K. Kharchenko, "Generalized identities with automorphisms" Algebra and Logic , 14 (1976) pp. 132–148 Algebra i Logika , 14 (1975) pp. 215–237
[a3]	V.K. Kharchenko, "Galois theory of semiprime rings" Algebra and Logic , 16 (1978) pp. 208–258 Algebra i Logika , 16 (1977) pp. 313–363
[a4]	V.K. Kharchenko, "Algebras of invariants of free algebras" Algebra and Logic , 17 (1979) pp. 316–321 Algebra i Logika , 17 (1978) pp. 478–487
[a5]	M. Lorenz, D.S. Passman, "Prime ideals in crossed products of finite groups" Israel J. Math. , 33 (1979) pp. 89–132
[a6]	M. Lorenz, D.S. Passman, "Prime ideals in group algebras of polycyclic-by-finite groups" Proc. London Math. Soc. , 43 (1981) pp. 520–543
[a7]	W.S. Martindale III, "The normal closure of the coproduct of rings over a division ring" Trans. Amer. Math. Soc. , 293 (1986) pp. 303–317
[a8]	W.S. Martindale III, S. Montgomery, "The normal closure of coproducts of domains" J. Algebra , 82 (1983) pp. 1–17
[a9]	S. Montgomery, "Outer automorphisms of semi-prime rings" J. London Math. Soc. , 18 : 2 (1978) pp. 209–220
[a10]	S. Montgomery, "-inner automorphisms of filtered algebras" Proc. Amer. Math. Soc. , 83 (1981) pp. 263–268
[a11]	S. Montgomery, D.S. Passman, "Crossed products over prime rings" Israel J. Math. , 31 (1978) pp. 224–256
[a12]	S. Montgomery, D.S. Passman, "-Inner automorphisms of group rings" Houston J. Math. , 7 (1981) pp. 395–402
[a13]	S. Montgomery, D.S. Passman, "-Inner automorphisms of group rings II" Houston J. Math. , 8 (1982) pp. 537–544
[a14]	S. Montgomery, D.S. Passman, "Galois theory of prime rings" J. Pure Appl. Algebra , 31 (1984) pp. 139–184
[a15]	J. Osterburg, D.S. Passman, "-inner automorphisms of enveloping rings" J. Algebra , 130 (1990) pp. 412–434