Matrix Algebras: Equivalent Ring Relations and Special Presentations

Recognizing when a ring is a matrix ring is of significant importance in the study of algebra. A well-known result in noncommutative ring theory states that a ring $R$ is a matrix ring if and only if it contains a set of $n\times n$ matrix units \{e_ij\}_i,j=1ⁿ; in which case R\cong M₂(S) for some $S$ that can be completely described in terms of these matrix units. However, finding and verifying a set of matrix units can be difficult. A more recent result states that a ring $R$ is an $(m+n)\times(m+n)$ matrix ring if, and only if, it contains three elements, $a$, $b$, and $f$, satisfying the two relations af^m+fⁿb=1 and f^m+n=0, in which case R\cong M_m+n(S) for some $S$. Under these relations very little is known about the structure of $S$. In this dissertation we investigate algebras over a commutative ring $A$ (or a field $k$) with elements $x$ and $y$ that satisfy the relations xⁱy+yx^j=1 and y²=0. We develop results about the structure of these algebras and their underlying rings when $\gcd(i,j)=1$ and then generalize these results for all $i$ and $j$. We then present some interesting examples demonstrating the more subtle characteristics of these algebras. Finally, we develop techniques to see when these algebras can be mapped to $2\times 2$ matrix rings.

Author

• Mendelson, Samuel Stephen

• Mathematics
• Diamond Lemma
• Free Algebras
• Matrix Recognition
• Matrix Relations
• Noncommutative Algebra

Identifier

hdl:1920/11246

OAI identifier oai:identifier

oai:MARS:1920/11246