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Let RRR be a ring with unity and Mod-RRR be the category of right RRR-modules. The Baer's Criterion for injectivity states that a right module MMM is injective iff it is RRR-injective, that is for each right ideal III of RRR, any homomorphism from III into MMM extends to RRR. Dually, a right module PPP is RRR-projective if for each right ideal III of RRR any homomorphism from MMM into R/IR/IR/I lifts to RRR. Unlike the case for injectivity, RRR-projective modules need not be projective. That is, the Dual Baer Criterion (DBC, for short) does not hold over every ring. The rings RRR for which the DBC holds in Mod-RRR 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 RRR-projectivity, and discuss some problems related to the rings whose injective right modules are RRR-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.