The family of rings the form
 \frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \rangle}
 is investigated which contains generalized Hamilton quaternions over $\Z_4$. These are local order 256. This has 256 contained in 88 distinct isomorphism classes. Of non-isomorphic rings, 10 minimal reversible nonsymmetric and 21 abelian reflexive nonsem...

Let [Formula: see text] be a commutative ring with text]. We introduce the concept of weakly 1-absorbing primary ideal, which is generalization ideal. A proper ideal said to if whenever nonunit elements and text], we have or number results concerning ideals are given, as well examples ideals. Furthermore, give corrected version result on rings.

We prove that plain, bounded, reversible and persistent Petri nets are weakly and strongly separable.