the aim of this short note is to introduce the concepts of prime and semiprime ideals in ordered ag-groupoids with left identity. these concepts are related to the concepts of quasi-prime and quasi-semiprime ideals, play an important role in studying the structure of ordered ag-groupoids, so it seems to be interesting to study them.

the generalized principal ideal theorem is one of the cornerstones of dimension theory for noetherian rings. for an r-module m, we identify certain submodules of m that play a role analogous to that of prime ideals in the ring r. using this definition, we extend the generalized principal ideal theorem to modules.

‎let $g$ be a group with identity $e.$ let $r$ be a $g$-graded‎ ‎commutative ring and $m$ a graded $r$-module‎. ‎in this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎for example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $m$ are studied‎.‎

let  be a graded ring and  be a graded -module. we define a topology on graded prime spectrum  of the graded -module  which is analogous to that for , and investigate several properties of the topology.

We frequently experience and successfully process anomalous utterances. Here we examine whether people do this by 'correcting' syntactic anomalies to yield well-formed representations. In two structural priming experiments, participants' syntactic choices in picture description were influenced as strongly by previously comprehended anomalous (missing-verb) prime sentences as by well-formed prim...

1. Let (a, b) and [a, b] be the greatest common divisor and the least common multiple of a and b, respectively . p„ denotes the n'th prime ; p, q, q1 , q2 , . . . are prime numbers. A sum Z and a product f f denote a summation and a multiplication, respectively, over primes indicated . The symbol # { . . .} denotes the number of elements indicated in the bracket 11. Pμ is the product of the fir...

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...

In this paper, we investigate the plasmonic near-field localization and the far-field scattering properties of non-periodic arrays of Ag nanoparticles generated by prime number sequences in two spatial dimensions. In particular, we demonstrate that the engineering of plasmonic arrays with large spectral flatness and particle density is necessary to achieve a high density of electromagnetic hot ...

The nature of unconscious information processing is a heavily debated issue in cognitive science (e.g., Kouider & Dehaene, 2007), and neuroscience (e.g., Crick & Koch, 1998). Traditionally, it has been thought that unconscious cognitive processing is restricted to knowledge that is strongly prepared by conscious processes (e.g., Dehaene et al., 1998). In three experiments, we show that the task...

An R-module M is called strongly noncosingular if it has no nonzero Rad-small (cosingular) homomorphic image in the sense of Harada. It is proven that (1) an R-module M is strongly noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring R is Artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...