Let G be a finite group and k a fixed algebraically closed field of characteristic p>0 . If p is odd, let H(; be the subring of H*(G,k) consisting of elements of even degree; following [20-22] we take H~=H*(G,k) if p=2, though one could just as well use the subring of elements of even degree for all p. H a is a finitely generated commutative k-algebra [13], and we let Va denote its associated a...

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>

In a previous paper [1], we defined the space of toric forms T (l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) 6=...

We show that every henselian valued field L of residue characteristic 0 admits a proper subfield K which is dense in L. We present conditions under which this can be taken such that L|K is transcendental and K is henselian. These results are of interest for the investigation of integer parts of ordered fields. We present examples of real closed fields which are larger than the quotient fields o...

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...

Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree  version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree  version of $C_c(X).$The main aim of this paper is to present t...

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.