The non-commutative full rhotrix ring and its subrings

Abstract

This paper presents the triple (R_n(F),+,^o) consisting of the set of all rhotrices of size n with entries in an arbitrary ring F; and together with the operations of rhotrix addition ‘+’ and row-column based method for rhotrix multiplication; 'o' , so as to introduce it as “the concept of non-commutative full rhotrix ring”, and study its properties. A number of subrings of (R_n(F),+,^o) are uncovered. Next, the paper shows that a particular subring of the non-commutative full rhotrix ring (R_n(F),+,^o)  is embedded in a particular subring of the well-known non-commutative full matrix ring (M_n(F),+,^.). Furthermore, isomorphic relationships between some subrings of (R_n(F),+,^o) are investigated.

Keywords: Ring, subring, Rhotrix Ring, Rhotrix Subring, Matrix Ring, Matrix Subring