### Abstract:

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 $\{e_{ij}\}_{i,j=1}^n$; in which case $R\cong M_2(S)$ 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 $af^m+f^nb=1$ 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 $x^iy+yx^j=1$ and $y^2=0$. 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.