On Rings with Involution and Inner Rings

Abstract

In this article, we extend several known results from the ring of all $n\times n$ matrices with complex entries to any ring $R$ with nonzero unity $1$ and involution $*$. We introduce various results concerning Hermitian, skew-Hermitian, Unitary and Normal elements of $R$. Also, we propose two versions of the norm of an element and the orthogonality of two elements of $R$. Furthermore, we define an order on the elements of $R$ and examine some properties. Finally, we establish the concept of inner rings and study some of its properties.