On weakly 1-absorbing primary submodules

Abstract

AboutPDF/EPUBRecommend To LibraryAbstractIn this paper, we introduce weakly 1-absorbing primary submodules of modules over commutative rings. LetR\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">RRbe a commutative ring with a nonzero identity andM\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">MMbe a nonzero unital module. A proper submoduleN\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">NNofM\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">MMis said to be a weakly 1-absorbing primary submodule if whenever0≠abm∈N\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">0≠abm∈N0≠abm∈Nfor some nonunit elementsa,b∈R\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">a,b∈Ra,b∈Randm∈M,\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">m∈M,m∈M,thenab∈(N:M)\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">ab∈(N:M)ab∈(N:M)orm∈M\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">m∈Mm∈M-rad(N),\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">rad(N),rad(N),whereM\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">MM-rad(N)\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">rad(N)rad(N)is the prime radical ofN.\" role=\"presentation\" style=\"display: inline; line-height: normal; font-size: 18px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\">N.N.Many properties and characterizations of weakly 1-absorbing primary submodules are given. We also give the relations between weakly 1-absorbing primary submodules and other classical submodules such as weakly prime, weakly primary, weakly 2-absorbing primary submodules. Also, we use them to characterize simple modules.