let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...

Let G be a finite group and let p prime number. We prove that if $$\chi \in {{{\text {Irr}}}}_{p'}(G)$$ $${\text {Ker}}\chi $$ does not have solvable normal p-complement then there exists $$\psi such (1)>\chi (1)$$ {Ker}}\psi <{\text . This is $$p'$$ -version of classical theorem Broline Garrison. As consequence, we obtain results on p-parts character codegrees.

We recall that a ring R is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal. Let R be a commutative ring, σ an automorphism of R and δ a σderivation of R. We recall that a prime ideal P of R is δ-divided if it is comparable (under inclusion) to every σ-invariant and δ-invariant ideal I (i.e. σ(I) ⊆ I and δ(I) ⊆ I) of R. A ring R is called a δ-divided ring...

In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...

It is shown that a commutative Bézout ring R with compact minimal prime spectrum is an elementary divisor ring if and only if so is R/L for each minimal prime ideal L. This result is obtained by using the quotient space pSpec R of the prime spectrum of the ring R modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if R is arith...

This paper exempliies how one can give constructive interpretations of proofs which uses the non-constructive principle that any ideal is contained in a prime ideal. We consider a textbook example AM69] which states that the coeecients of an invertible polynomial in a commutative ring with unit are nilpotent. This statement has a simple non-constructive proof which relies on the existence of ce...

In [17] Lee and Shiue showed that if R is a non-commutative prime ring, I a nonzero left ideal of R and d is a derivation of R such that [d(x)x, x]k = 0 for all x ∈ I, where k,m, n, r are fixed positive integers, then d = 0 unless R ∼= M2(GF (2)). Later in [1] Argaç and Demir proved the following result: Let R be a non-commutative prime ring, I a nonzero left ideal of R and k,m, n, r fixed posi...

The $J/{\psi}$ weak decays are rare but possible within the standard model of elementary particles. Inspired by potential prospects at future intensity frontier, $C$ parity violating ${\to}$ ${\pi}{\eta}^{({\prime})}$, ${\eta}{\eta}^{\prime}$ and strangeness changing ${\pi}K$, $K{\eta}^{({\prime})}$ studied with perturbative QCD approach. It is found that have relatively large branching ratios,...

We investigate the structure of prime ideals of finite height in polynomial extension rings of a commutative unitary ring R. We consider the question of finite generation of such prime ideals. The valuative dimension of prime ideals of R plays an important role in our considerations. If X is an infinite set of indeterminates over R, we prove that every prime ideal of R[X] of finite height is fi...

Let N ≡ 1 mod 4 be the negative of a prime, K = Q( √ N) and OK its ring of integers. Let D be a prime ideal in OK of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters ψD of K with conductor (D) and infinite type (1, 0). Their L-series L(ψD , s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspu...