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
The notion of a Euclidean module over a commutative ring
as a natural extension of the Euclidean rings was studied in .
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