In this paper, we construct subring homomorphic encryption scheme that is a homomorphic encryption scheme built on the decomposition ring, which is a subring of cyclotomic ring. In the scheme, each plaintext slot contains an integer in Zpl , rather than an element of GF(p) as in conventional homomorphic encryption schemes on cyclotomic rings. Our benchmark results indicate that the subring homo...

A minimum depth is assigned to a ring homomorphism and a bimodule over its codomain. When the homomorphism is an inclusion and the bimodule is the codomain, the recent notion of depth of a subring in a paper by Boltje-Danz-Külshammer is recovered . Subring depth below an ideal gives a lower bound for BDK’s subring depth of a group algebra pair or a semisimple complex algebra pair.

1-2. (a) Let u, v ∈ S and suppose that uv = 0; then u = 0 or v = 0 since u, v ∈ R and R is an integral domain. Consider the unit homomorphisms η : Z −→ R and η′ : Z −→ S. Then for n ∈ Z, η′(n) = η(n), so ker η′ = ker η and therefore charS = charR. (b) Q is a field and Z ⊆ Q is a subring which is not a field. 1-3. (a) For any subring R ⊆ C, R is an integral domain with characteristic subring Z a...

<p>In this paper, we define the - -fuzzy subring and discussed various fundamental aspects of subrings. We introduce concept -level subset new fuzzy set prove that form a ring. ideal show all cosets Moreover, investigate properties homomorphic image subring.</p>

Remark (Ideals versus subrings): It is worthwhile to compare these two notions; they are related, but with subtle and important differences. Both an ideal I and a subring S of a ring R are subsets of R which are subgroups under addition and are stable under multiplication. However, each has an additional property: for an ideal it is the absorption property (IR2). For instance, the integers Z ar...

1. Throughout this paper, K will represent a division ring and L a galois division subring. We are interested in establishing a galois theory for the extension K/L when K/L is locally finite. In order to do this one must identify the galois subrings of K containing L. An example given by Jacobson [4] shows that not every such division subring is galois. However, we obtain that each subring subj...

We give new proofs of two theorems on rings in which every zero subring is finite; and we apply these theorems to obtain a necessary and sufficient condition for an infinite ring with periodic additive group to have an infinite periodic subring. 2000 Mathematics Subject Classification. 16N40, 16N60, 16P99. Let R be a ring and N its set of nilpotent elements; and call R reduced if N = {0}. Follo...

We study semisimple Hopf algebra actions on Artin–Schelter regular algebras and prove several upper bounds the degrees of minimal generators invariant subring, syzygies modules over subring. These results are analogues for group commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, Symonds.