CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

In noncommutative ring theory identities of the form $axb=bxa$, for all $x$ in some noncommutative ring, have played an important role. If the ring in question is a division ring $D$ (that is, a ring in which every non zero element is invertible), it is easy to show that the identity implies $a$ and $b$ are linearly independent over the center of $D$. An important result in ring theory was extending results of this nature to a class or rings called prime rings. We further generalized this result to semi prime rings, by studying identities of the form $axb=c\sigma \left(x\right)d$, $\sigma$ an automorphism of the ring. In the talk, we will discuss these issues over prime rings. A prime ring is one in which the product of two nonzero ideals is nonzero (for example, $M_n\left(\mathbb{Z}\right)$ , the ring of $n$ by $n$ matrices over the integers). A semi prime ring is one in which there are no nonzero nilpotent ideals (so, prime implies semi prime). One difficulty in studying prime rings is that their centers are in general not fields. Given a prime ring $R$, it was discovered in the 1960s that there is a way of constructing a type of quotient ring $Q\left(R\right)$ of $R$, which is also prime, and whose center $C$ is a field. Closely related to $R$ is the subring $RC$ of $Q\left(R\right)$ is called the central closure of $R$ and it is a prime algebra over $C$. The results on identities are established using this construction. We will show how this can be accomplished over prime rings.