George M. Bergman
Bilinear maps on Artinian modules.
Mon, 08/15/2011 - 15:01 — SomNambulAuthors: George M. Bergman
Subjects: Rings and Algebras
AbstractIt is shown that if a bilinear map f: A x B --> C of modules over a
commutative ring k is nondegenerate (i.e., if no nonzero element of A
annihilates all of B, and vice versa), and A and B are Artinian, then A and B
are of finite length.Some consequences are noted. Counterexamples are given to some attempts to
generalize the above statement to balanced bilinear maps of bimodules over
noncommutative rings, while the question is raised whether other such
generalizations are true.An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange.
Tue, 01/12/2010 - 16:01 — SomNambulAuthors: George M. Bergman
Subjects: Rings and Algebras
AbstractThe inner automorphisms of a group G can be characterized within the category
of groups without reference to group elements: they are precisely those
automorphisms of G that can be extended, in a functorial manner, to all groups
H given with homomorphisms G --> H. Unlike the group of inner automorphisms of
G itself, the group of such extended systems of automorphisms is always
isomorphic to G.Homomorphic images of pro-nilpotent algebras.
Wed, 12/02/2009 - 16:01 — SomNambulAuthors: George M. Bergman
Subjects: Rings and Algebras
AbstractIt is shown that any finite-dimensional homomorphic image of an inverse limit
of nilpotent not-necessarily-associative algebras over a field is nilpotent.
More generally, this is true of algebras over a general commutative ring k,
with "finite-dimensional" replaced by "of finite length as a k-module".Linear maps on k^I, and homomorphic images of infinite direct product algebras.
Wed, 10/28/2009 - 16:01 — SomNambulAuthors: George M. Bergman, and Nazih Nahlus
Subjects: Rings and Algebras
AbstractLet k be an infinite field, I an infinite set, V a k-vector-space, and
g:k^I\to V a k-linear map. It is shown that if dim_k(V) is not too large (under
various hypotheses on card(k) and card(I), if it is finite, respectively
countable, respectively < card(k)), then ker(g) must contain elements
(u_i)_{i\in I} with all but finitely many components u_i nonzero.Homomorphisms on infinite direct product algebras, especially Lie algebras.
Wed, 10/28/2009 - 16:01 — SomNambulAuthors: George M. Bergman, Nazih Nahlus
Subjects: Rings and Algebras
AbstractWe study surjective homomorphisms f:\prod_I A_i\to B of
not-necessarily-associative algebras over a commutative ring k, for I a
generally infinite set; especially when k is a field and B is
countable-dimensional over k.Our results have the following consequences when k is an infinite field, the
algebras are Lie algebras, and B is finite-dimensional:If all the Lie algebras A_i are solvable, then so is B.
If all the Lie algebras A_i are nilpotent, then so is B.
On coproducts in varieties, quasivarieties and prevarieties.
Mon, 08/17/2009 - 18:30 — SomNambulAuthors: George M. Bergman
Subjects: Rings and Algebras
AbstractIf the free algebra F on one generator in a variety V of algebras (in the
sense of universal algebra) has a subalgebra free on two generators, must it
also have a subalgebra free on three generators? In general, no; but yes if F
generates the variety V.
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