Prime structures in a Morita context
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ORIGINAL ARTICLE
Prime structures in a Morita context Mete Burak Calci1 • Sait Halicioglu2 • Abdullah Harmanci3 • Burcu Ungor2 Received: 3 July 2019 / Accepted: 28 April 2020 Sociedad Matemática Mexicana 2020
Abstract In this paper, we study the primeness and semiprimeness of a Morita context. Necessary and sufficient conditions are investigated for an ideal of a Morita context to be a prime ideal and a semiprime ideal. In particular, we determine the conditions under which a Morita context is prime and semiprime. Keywords Morita context Prime ideal Semiprime ideal Prime radical Prime ring Semiprime ring
Mathematics Subject Classification 16D80 16S50 16S99
1 Introduction Throughout this paper, all rings are associative with 1 6¼ 0 and modules are unitary. A proper ideal I of a ring R is called prime if for any elements a and b in R, aRb I implies a 2 I or b 2 I, and equivalently if for every pair of ideals A, B of R, AB I implies A I or B I. The prime radical of a ring R is the intersection of all prime ideals of R and denoted by P(R). It is clear that P(R) contains every nilpotent ideal of R. Recall that a proper ideal I of R is said to be semiprime if for any a 2 R, & Mete Burak Calci [email protected] Sait Halicioglu [email protected] Abdullah Harmanci [email protected] Burcu Ungor [email protected] 1
Tubitak, Bilgem, Gebze, Kocaeli, Turkey
2
Department of Mathematics, Ankara University, Ankara, Turkey
3
Department of Mathematics, Hacettepe University, Ankara, Turkey
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aRa I implies a 2 I, and equivalently for every ideal A of R, A2 I implies A I. The concepts of prime and semiprime ideals have very important roles in ring theory. A ring R is called prime if 0 is a prime ideal, and equivalently for any a; b 2 R, aRb ¼ 0 implies a ¼ 0 or b ¼ 0. A ring R is said to be semiprime if it has no nonzero nilpotent ideals, and equivalently for any a 2 R, aRa ¼ 0 implies a ¼ 0. Obviously, prime rings are semiprime. Also, it is well known that a ring R is semiprime if and only if PðRÞ ¼ 0. A proper submodule N of a left R-module M is called prime if for any r 2 R and m 2 M, rRm N implies rM N or m 2 N. It is easy to show that if N is a prime submodule of M, then the annihilator of the module M/N is a two-sided prime ideal of R. A module M is said to be prime if 0 is a prime submodule of M, i.e., for any r 2 R and m 2 M, rRm ¼ 0 implies rM ¼ 0 or m ¼ 0. A Morita context is a 4-tuple (R, V, W, S), where R, S are rings, R VS and S WR are bimodules with context products V W ! Rand W V ! S written multiplicaR V tively as ðv; wÞ7!vw and ðw; vÞ7!wv, such that is an associative ring with W S the usual matrix operations. Morita contexts appeared in the work of Morita [6] and they play an important role in the category of rings and modules to describe equivalences between full categories of modules over rings. One of the fundamental results in this direction says that the categories of left modules over the rings R and S are equivalent if
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