Theorem.2.1 Let R and S be finite

commutative rings. Then [(R x S.sup.)x] = [R.sup.x] x [S.sup.x] as groups.

The 14 workshop notes, plenary talks, and invited papers look at representations of algebras from such perspectives as polyhedral models for tensor product multiplicities, Nakayama-type phenomena in higher Auslander-Reiten theory, finite dimensional algebras arising as blocks of finite group algebras, thick tensor ideals of right bounded derived categories of

commutative rings, and computations and applications of some homological constants for polynomial representations of GLn.

The most remote sources of partial actions of Hopf algebras can be found in the theory of partial Galois extensions, which was a generalization of the Galois theory for

commutative rings by Chase, Harrison and Rosenberg to the case of partial group actions.

This article investigates the weakly completely 2-absorbing primary fuzzy ideal and 2-absorbing primary fuzzy ideal as a generalization of primary fuzzy ideal in

commutative rings. Also some characterizations of 2-absorbing primary fuzzy ideal are obtained.

Recurrence Formulas for n-Symmetric Sets of

Commutative Rings with Unity.

Brown, Matrices Over

Commutative Rings, Marcel Dekker, New York, NY, USA, 1993.

Darani, "Generalizations of primal ideals in

commutative rings," Matematichki Vesnik, vol.

The idea of constructing FHE schemes from

commutative rings was initially demonstrated by Kipnis and Hibshoosh [61] in 2012.

One of the earliest realizations of [F.sub.1] was through the scheme theory developed by Deitmar [1], which is based on the observation that

commutative rings over [F.sub.1] could be imagined as commutative multiplicative monoids (with an absorbing element).

The divisibility theory of

commutative rings is a fundamental and persisting topic in mathematics that entails two main aspects: determining irreducibility and finding a factorization of the reducible elements in the ring.

All rings considered in this paper will be

commutative rings with identity.

Smith defined weakly prime ideals in

commutative rings, an ideal P of a ring R is weakly prime if 0 [not equal to] ab [member of] P implies a [member of] P or b [member of] P.