let $r$ be a commutative ring with identity and $m$ be a unitary$r$-module. the primary-like spectrum $spec_l(m)$ is thecollection of all primary-like submodules $q$ such that $m/q$ is aprimeful $r$-module. here, $m$ is defined to be rsp if $rad(q)$ isa prime submodule for all $qin spec_l(m)$. this class containsthe family of multiplication modules properly. the purpose of thispaper is to intro...

Let $R$ be a commutative ring with identity and $M$ be a unitary$R$-module. The primary-like spectrum $Spec_L(M)$ is thecollection of all primary-like submodules $Q$ such that $M/Q$ is aprimeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ isa prime submodule for all $Qin Spec_L(M)$. This class containsthe family of multiplication modules properly. The purpose of thispaper is to intro...

In this paper we aim at the description of foliations having tangent sheaf TF with c1(TF) = c2(TF) = 0 on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of F is an Abelian variety. It turns out that the analytic type of the Zariski closures of leaves may vary from leaf to leaf. W...

Let a, b, c, d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in P defined by ax + by + cz + dw = 0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic

This paper studies algebraic frames L and the set Min(L) of minimal prime elements of L. We will endow the set Min(L) with two well-known topologies, known as the Hullkernel (or Zariski) topology and the inverse topology, and discuss several properties of these two spaces. It will be shown that Min(L) endowed with the Hull-kernel topology is a zero-dimensional, Hausdorff space; whereas, Min(L) ...

We prove a product decomposition of the Zariski closure of the jet lifts of an entire curve f : C → A into a semi-abelian variety A, provided that f is of finite order. On the other hand, by giving an example of f into a three dimensional abelian variety we show that this product decomposition does not hold in general; there was a gap in the proofs of [2], Proposition 1.8 (ii) and of [6], Theor...

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups G defined over an algebraically closed field k of characteristic 0. That is, if Φ: G −→ G is a dominant endomorphism, we prove that one of the following holds: either there exists a non-constant rational function f ∈ k(G) preserved by Φ (i.e., f ◦ Φ = f), or there exists a point x ∈ ...

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups G defined over an algebraically closed field k of characteristic 0. That is, if Φ: G −→ G is a dominant endomorphism, we prove that one of the following holds: either there exists a non-constant rational function f ∈ k(G) preserved by Φ (i.e., f ◦ Φ = f), or there exists a point x ∈ ...

<abstract><p>Let $ R be a G graded commutative ring and M $-graded $-module. The set of all second submodules is denoted by Spec_G^s(M), it called the spectrum $. We discuss rings with Noetherian prime spectrum. In addition, we introduce notion Zariski socle explore their properties. also investigate Spec^s_G(M) topology from viewpoint being space.</p></abstract>

Let A" be a Lie subgroup of the connected, simply connected nilpotent Lie group G, and let f, g be the corresponding Lie algebras. Suppose that a is an irreducible unitary representation of K. We give an explicit direct integral decomposition of IndA^c;o into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between GA and the coadjoint orbits in g* (and sim...