The non-commutative full rhotrix ring and its subrings

This paper presents the triple (Rn(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 (Rn(F),+,o) are uncovered. Next, the paper shows that a particular subring of the non-commutative full rhotrix ring (Rn(F),+,o) is embedded in a particular subring of the well-known non-commutative full matrix ring (Mn(F),+,.). Furthermore, isomorphic relationships between some subrings of (Rn(F),+,o) are investigated.