A

commutative ring is a ring in which the multiplicative operation is commutative.

R is a

commutative ring with unity 1 [??] N(R, I) is a commutative Neutrosophic ring with unity 1 and Neutrosophic unity I.

Let A and B be two unital algebras over a

commutative ring R, where R is a unital

commutative ring, and let M be a unital (A, B)--bimodule.

[36] Let k be a

commutative ring and A and B be k-algebras and [alpha] is a unital partial action of a finite group G on A.

Let k be a

commutative ring and M be a g-bimodule of an associative (not necessarily commutative) k-algebra g.

Recall that if I and J are ideals of a

commutative ring R, then their ideal quotient denotes (I : J) defined by (I: J) = {r [member of] R | rJ [subset] I}.

Prime ideals and primary ideals play a significant role in

commutative ring theory.

Let R = (R, +, *) be a

commutative ring of characteristic m > 0 with unity e and zero 0.

Let G be an abelian group and let R be any

commutative ring with unity; then R is called a G-graded ring (for short graded ring), if R = [[direct sum].sub.g[member of]G][R.sub.g] such that if a,b [member of] G, then [R.sub.a][R.sub.b] =[subset or equal to] [R.sub.ab].

Note that, as is mentioned in Introduction, for a square matrix A whose components lie in a

commutative ring R, the form 1/det(1-uA) can always be reformulated in a generating function of exponential type, that is, if we let [N.sub.m] = tr [A.sup.m] for each m [member of] [Z.sub.[greater than or equal to]1], then the form equals

Instead of using the lattice, these Noise-free FHE schemes are constructed based on the classical number-theoretic concepts such as octonion algebra,

commutative ring, and non-commutative ring.

Matsumura

Commutative Ring Theory Cambridge tudies in Advanced Mathematics 8 Cambridge University Press(1986).