In this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. We prove that if $N$ is a weakly prime subtractive ternary subsemimodule of a ternary $R$-semimodule $M$, then either $N$ is a prime ternary subsemimodule or $(N : M)(N : M)N = 0$. If $N$ is a $Q$-...

Let X be a smooth hypersurface of dimension n ≥ 1 and degree d 3 in the projective space given as zero set homogeneous form F. If (n, d) ≠ (1, 3), (2, 4) it is well known that every automorphism extends to an space, i.e., Aut(X) ⊆ PGL(n + 2, ℂ). We say group liftable if there exists subgroup GL(n ℂ) projecting isomorphically onto leaving F invariant. Our main result this paper shows only 2 are ...

We study the real components of modular curves. Our main result is an abstract group-theoretic description of the real components of a modular curve defined by a congruence subgroup of level N in terms of the corresponding subgroup of SL2(Z/NZ). We apply this result to several families of modular curves (such as X0(N), X1(N), etc.) to obtain formulas for the number of real components. Somewhat ...

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 2 P...

Let A be an elliptic curve over Q of square free conductor N . Suppose A has a rational torsion point of prime order r such that r does not divide 6N . We prove that then r divides the order of the cuspidal subgroup C of J0(N). If A is optimal, then viewing A as an abelian subvariety of J0(N), our proof shows more precisely that r divides the order of A ∩ C. Also, under the hypotheses above, we...

Given a $p$-group $G$ and subgroup-closed class $\mathfrak{X}$, we associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count $\mathfrak{X}$-subgroups containing subject to further properties. We show in Theorem I that one of the said is always $\equiv 1 \pmod p$ if only same holds for others. In II supplement above result by focusing on normal III obtain sharpened version c...

1. Introduction. Let p be a prime number and F an algebraic closure of the finite field F p with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations ρ of the Galois group G Q = Gal(¯ Q/Q) into GL(n, F), unramified at all finite primes not dividing pN. (We shall say ρ is unramified outside pN.) In this paper, representation will always mean contin...

For each odd prime p, we prove the existence of infinitely many real quadratic fields which are p-rational. Explicit imaginary and bi-quadratic p-rational also given for p. Using a recent method developed by Greenberg, deduce Galois extensions Q with group isomorphic to an open subgroup GLn(Zp), n=4 n=5 at least all primes p<192.699.943.