In this paper we exhibit some examples of non-Gorenstein Hecke algebras, and hence some modular forms for which mod 2 multiplicity one does not hold. Define S2(Γ0(N)) to be the space of classical cuspidal modular forms of weight 2, level N , and trivial character. The Hecke algebra TN is defined to be the subring of End(S2(Γ0(N)) generated by the Hecke operators {Tp : p 6 |N} and {Uq : q|N}. Le...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

Let J (M) denote the σ-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of J (M) is bigger than the covering number of the ideal of the meager subsets of ω. We also show that Martin’s Axiom implies that the additivity of J (M) is 2. Finally we prove that there are no analytic infinite maximal antichains in any finite pro...

let d be a division ring with centre k and dim, d< ? a valuation on k and v a noninvariant extension of ? to d. we define the initial ramfication index of v over ?, ?(v/ ?) .let a be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to d with valuation rings a , a ,…, a . if b= a , it is shown that the following conditions are equivalent: (i) b is a finite...

‎in this paper‎, ‎we introduce the notion of the radical of a $pmv$-algebra $a$ and we charactrize radical $a$ via elements of $a$‎. ‎also‎, ‎we introduce the notion of the radical of a $cdot$-ideal in $pmv$-algebras‎. ‎several characterizations of this radical is given‎. ‎we define the notion of a semimaximal $cdot$-ideal in a $pmv$-algebra‎. ‎finally we show that $a/i$ has no nilpotent elemen...

Let R be a commutative ring with unity, M unitary R-module and G finite abelian group (viewed as \({\mathbb {Z}}\)-module). The main objective of this paper is to study properties mod-annihilators M. For \(x \in M\), we the ideals \([x : M] =\{r\in ~|~ rM\subseteq Rx\}\) corresponding mod-annihilator We investigate when [x an essential ideal R. prove that arbitrary intersection represented by i...

Let D be a division ring with centre K and dim, D< ? a valuation on K and v a noninvariant extension of ? to D. We define the initial ramfication index of v over ?, ?(v/ ?) .Let A be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to D with valuation rings A , A ,…, A . If B= A , it is shown that the following conditions are equivalent: (i) B i...

Let k be an algebraic number eld with maximal order o k and let m 0 be an ideal of o k. We develop an algorithm for the computation of the multiplicative group (o k =m 0) of the residue class ring o k =m 0. It is applied to the computation of the ray class group Cl m k modulo m = m 0 m 1 , where m 1 denotes a formal product of real innnite places, and also to the computation of the conductor of...

It is proved that if A is aregular uniform algebra on a compact Hausdorff space X in which every closed ideal is an M-ideal, then A = C(X).

‎In this paper‎, ‎we introduce the notion of the radical of a $PMV$-algebra $A$ and we charactrize radical $A$ via elements of $A$‎. ‎Also‎, ‎we introduce the notion of the radical of a $cdot$-ideal in $PMV$-algebras‎. ‎Several characterizations of this radical is given‎. ‎We define the notion of a semimaximal $cdot$-ideal in a $PMV$-algebra‎. ‎Finally we show that $A/I$ has no nilpotent elemen...