Given a

commutative ring k and a unital partial action [alpha], of a group G on a k-algebra A, we define the partial invariant subalgebra as

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.

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).

The numbers 0 through 10 form the cyclic group or

commutative ring upon which summation (mod 11) and multiplication (mod 11) are defined.

ij]] is an n by n matrix over a

commutative ring, then the standard determinant expression of A is

1 Let B be a finite left regular band, k a

commutative ring with unit, and X,Y G A (B).

Foster, The idempotent elements of a

commutative ring form a Boolean Algebra, Duke math.

Note that A is a faithful left-module over End A, and therefore it is a faithful module over the

commutative ring R.

The notion of a Euclidean module over a

commutative ring as a natural extension of the Euclidean rings was studied in [4].

Recall that the annihilator of an element r in a

commutative ring R is the collection, denoted ann(r), of elements s in R with the property that rs = 0.

This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any

commutative ring.