[Turkmath:4908] Seminar on May 28, 2021 at 16:00 via Zoom

[Mehmet Akif Erdal]

Dear list members,

You are most cordially invited to the Yeditepe Mathematics Department 25th
Year Seminars organized by the Department of Mathematics. The details of
this week's talk are as follows (please note the unusual time):

Speaker: Engin Büyükaşık (İzmir Institute of Technology)

Title: Dual Baer Criterion and R-projectivity of injective modules

Abstract:  Let $R$ be a ring with unity and Mod-$R$ be the category of
right $R$-modules. The Baer's Criterion for injectivity states that a right
module $M$ is injective iff it is $R$-injective, that is  for each right
ideal $I$ of $R$, any homomorphism from $I$ into $M$ extends to $R$.
Dually, a right module $P$ is $R$-projective if for each right ideal $I$ of
$R$ any homomorphism  from $M$ into $R/I$ lifts to $R$.   Unlike the case
for injectivity, $R$-projective modules need not be projective. That is,
the Dual Baer Criterion (DBC, for short) does not hold over every ring. The
rings $R$ for which the DBC holds  in Mod-$R$ are called right testing.
>From [4], it is known that right perfect rings are right testing. In  [3],
 Faith stated the characterization of all right testing rings as an open
problem. Recently in [6], Trlifaj proved that the problem of characterizing
right testing rings is undecidable in ZFC.

In this talk, after summarizing the aforementioned results, I will mention
an extend of  the notion of $R$-projectivity, and discuss some problems
related to the rings whose injective right modules are $R$-projective which
are partially solved in [1].

References
[1] Y. Alagöz and E. Büyükaşık, Max-projective modules, J. Algebra
Appl. 20 (2021),
no. 6. 2150095.
[2] H. Alhilali, Y. Ibrahim, G. Puninski, and M. Yousif, When R is a
testing module for projectivity?
J. Algebra 484 (2017), 198-206.
[3] C. Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976. Ring
theory, Grundlehren der Mathematischen Wissenschaften, No. 191.
[4] F .L. Sandomierski, Relative injectivity and projectivity, 1964. Thesis
(Ph.D.) The Pennsylvania
State University.
[5] J. Trlifaj, Whitehead test modules, Trans. Amer. Math. Soc. 348 (1996),
no. 4, 1521-1554.
[6] J. Trlifaj, Faith’s problem on R-projectivity is undecidable, Proc.
Amer. Math. Soc. 147 (2019),
no. 2, 497-504.
[7] J. Trlifaj, The dual Baer Criterion for non-perfect rings, Forum Math. 32
(2020), no. 3, 663-672.

Date: Friday, May 28, 2021
Time: 16:00
Zoom: Meeting ID: 889 3945 0567
Passcode: 7tpSeminar


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