Properties of Homomorphisms and Centers in Multiset Rings.

Multiset rings (Mrings) form an important generalization of classical rings by allowing elements to appear with finite frequencies and frequency function satisfies some conditions. In non-commutative Mrings, the center (called Madhya, denoted by , where Ƴ is an Mring) consists of those elements that frequency-commute with every element of the ring with respect to multiplication. This paper studies the algebraic behavior of the center of an Mring constructed over a ring. We prove that if is a non-empty Mring over a ring (S, +, ⋅) which is also form a field and satisfies the natural condition that every non-zero element has the same frequency as its multiplicative inverse, i.e., (u) = (u⁻¹) for all u ≠ 0, then the center ( ) itself becomes a sub-Mring of Ƴ. This result shows that the center inherits the full Mring structure in the presence of multiplicative inverses and frequency symmetry. Additionally, we establish a homomorphism theorem for Mideals and centers: if ɧ: (S, ⊕, ⊙) → (T, +, ⋅) is a bijective Mring homomorphism, then the pre-image of every Mideal of T is an Mideal of S, and the centers satisfy a natural correspondence, highlighting when the center coincides with the whole Mring and when it is properly smaller. These results extend classical center and correspondence theorems to the multiset setting and open the door to further investigation of representation theory, idempotent structures, and applications in coding theory and cryptographic protocols based on multiplicity-aware algebraic systems.