On Amenable and Congenial Bases for Infinite Dimensional Algebras

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" On Amenable and Congenial Bases for Infinite Dimensional Algebras "
Muhammad, Rebin A. López-Permouth, Sergio R.

Document Type	:	Latin Dissertation
Language of Document	:	English
Record Number	:	1107305
Doc. No	:	TLpq2440081481
Main Entry	:	López-Permouth, Sergio R.
	:	Muhammad, Rebin A.
Title & Author	:	On Amenable and Congenial Bases for Infinite Dimensional Algebras\ Muhammad, Rebin A.López-Permouth, Sergio R.
College	:	Ohio University
Date	:	2020
student score	:	2020
Degree	:	Ph.D.
Page No	:	74
Abstract	:	The study of the recently introduced notions of amenability, congeniality and simplicity of bases for infinite dimensional algebras is furthered. A basis B over an infinite dimensional F-algebra A is called amenable if FB, the direct product indexed by B of copies of the field F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules. If B is congenial to C but C is not congenial to B, then we say that B is properly congenial to C. An amenable basis B is called simple if it is not properly congenial to any other amenable basis and it is called projective if there does not exist any amenable basis which is properly congenial to B. We introduce a few families of algebras and study these notions in those contexts; in particular, we show the families to include examples of algebras without simple or projective bases. Our examples will illustrate the one-sided nature of amenability and simplicity as they include examples of bases which are amenable only on one side and, likewise, bases which have only one one-sided simple. We also consider the notions of amenability and congeniality of bases for infinite dimensional algebras in the context of the tensor product of bases for tensor products of algebras. In particular we give conditions for tensor product algebras to have simple bases in terms of the properties of the algebras being combined. Our results are then used to extend earlier results on the existence of simple bases in the algebra of single-variable polynomials to algebras of polynomials on several variables.
Subject	:	Mathematics