[Paper Review] Doing Algebra over an Associative Algebra

Nathan BeDell|arXiv (Cornell University)|Aug 3, 2017

Advanced Topics in Algebra5 references3 citations

TL;DR

This paper explores elementary algebraic properties—such as factorization, divisibility, and zero divisors—within finite-dimensional unital associative algebras over ℝ, using a hands-on, accessible approach akin to high school algebra. It introduces the nil poset, a novel order-theoretic structure derived from multiplicative bases, to systematically analyze annihilators and zero divisors in nilpotent algebras, offering a foundational tool for extending calculus and differential equations to non-field algebras.

ABSTRACT

A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to $\mathbb{R}$ and imposing the relation $i^2 = -1$. In this paper, we examine some of the elementary algebraic properties of such algebras, how they break-down when compared to standard grade-school algebra, and discuss how such properties are relevant to other areas of our research regarding algebras, such as the $\mathcal{A}$-calculus and the theory of $\mathcal{A}$-ODEs.

Motivation & Objective

• To investigate how standard algebraic principles from grade-school algebra—like factorization and divisibility—behave in finite-dimensional unital associative algebras over ℝ.
• To identify and characterize the breakdown of classical algebraic properties due to the presence of zero divisors and nilpotent elements.
• To develop an accessible, elementary framework for understanding algebraic structures in non-field settings, suitable for undergraduates and early researchers.
• To introduce and formalize the nil poset as a tool for analyzing annihilators and structural properties of nil algebras.
• To lay groundwork for extending calculus and differential equations to algebras, by first understanding their underlying algebraic behavior.

Proposed method

• Uses the finite-dimensional vector space structure of algebras to define and compute matrix representations (regular representation) of algebra elements via left multiplication maps.
• Applies basis-dependent matrix representations $ M_{eta}( heta) $ to model algebraic operations and preserve algebraic structure isomorphically.
• Introduces the concept of a multiplicative basis, where basis elements multiply to scalar multiples of other basis elements, enabling systematic analysis.
• Defines the nil poset $ ewcommand{ ilposet}{ ext{NilPoset}} ilposet(eta) $ as a partially ordered set on basis elements, with $ v_i \preceq v_j $ if $ v_i \star v_k = c v_j $ for some $ v_k $, $ c \in \mathbb{R} $.
• Uses Hasse diagrams of the nil poset to visually trace paths that yield annihilating elements, such as $ \epsilon^n \gamma^m $, to compute annihilators of basis elements.
• Applies the nil poset to analyze annihilator ideals, e.g., reading $ \mathrm{Ann}(\epsilon\gamma) = \mathrm{span}\{\epsilon^2, \gamma^2, \epsilon\gamma\} $ from the diagram.

Experimental results

Research questions

• RQ1 How do classical algebraic properties like divisibility and unique factorization fail in associative algebras with zero divisors and nilpotents?
• RQ2 What is the structural role of the nil poset in characterizing annihilators of basis elements in unital nil algebras?
• RQ3 Can any poset be realized as the nil poset of some finite-dimensional unital nil algebra?
• RQ4 How do order-theoretic properties of the nil poset—such as being a lattice, modular, or distributive—relate to algebraic properties of the underlying algebra?
• RQ5 What is the minimal algebraic structure required to support a given nil poset, and what constraints does this impose?

Key findings

• The nil poset $ \nilposet(\mathcal{A}) $, defined via covering relations from left multiplication, provides a visual and computational method to determine annihilators of basis elements in multiplicative nil algebras.
• For the algebra $ \Gamma_3 \otimes \Gamma_3 $, the Hasse diagram shows that $ \mathrm{Ann}(\epsilon\gamma) = \mathrm{span}\{\epsilon^2, \gamma^2, \epsilon\gamma\} $, recoverable by tracing paths from $ \epsilon\gamma $ to 0.
• The nil poset is always a poset under the defined ordering $ v_i \preceq v_j $, and its structure reflects the multiplicative closure of the basis.
• Not all nil posets are lattices; a counterexample is provided using a 6-generator algebra with non-unique minimal upper bounds, showing $ \epsilon $ and $ \gamma $ have two distinct minimal upper bounds $ \zeta $ and $ \xi $.
• The method of path tracing in the Hasse diagram allows one to generate annihilating monomials $ \epsilon^n \gamma^m $, with $ n $ and $ m $ counting left and right steps, respectively.
• The nil poset construction is general and can be applied to any unital nil algebra with a multiplicative basis, offering a systematic, visual alternative to abstract ideal-theoretic analysis.