A note on the countable union of prime submodules

Let M be a finitely-generated module over a Noetherian ring R. Suppose is an ideal of R and let N=M and =Ann(M/N). If ⫅J(R), M is complete with respect to the -adic topology, {Pi}i≥1 is a countable family of prime submodules of M, and x∈M, then x+N⫅∪i≥1Pi implies that x+N⫅Pj for some i≥1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules

Zero-divisor graph with respect to an ideal

All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately

Elements with square roots in finite groups

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups. </jats:p

La Poliomyelite en Iran Premieres Recherches de Laboratoire

This research work on the status of poliomyelitis in Iran, performed in 1954, in 1958-59, consisted in providing data on the past history of the infection and on its present diffusion in this country. The serological and COprologi~al surveys produced evidence of conditions similar to thos.e of other countries .WIth the same public health situation: large and permanent (WIthout seasonal v~riatlOnSJ diffusion of the e~teroviruses, reaching percentages of 37 to 74% (accol~&apos;dmg to the groups) of ca:rri!ers-excreters in a sampling of healthy chi~dren u~der five y~a!&apos;s of age, providing high levels of early and complete Immunlzation against the pohomyelitic disease