RD-projective module whose subprojectivity domain is minimal

Abstract

A p-indigent module is one that is subprojective only to projective modules. An RD-projective module is subprojective to any torsionfree (and flat) module. An RD-projective module $T$ is called rdp-indigent if it is subprojective only to torsionfree modules. In this work, we consider the structure of SRDP rings whose (simple) RD-projective right $R$-modules are rdp-indigent or torsionfree. Moreover, new characterizations of P-coherent rings and torsionfree rings are presented by subprojectivity domains.

Keywords

• RD-projective module
• subprojectivity domain
• rdp-indigent module
• torsionfree module
• flat module

References

1. [1] Y. Alagöz and Y. Durğun, An alternative perspective on pure-projectivity of modules, São Paulo J. Math. Sci. 14 (2), 631–650, 2020.
2. [2] A.N. Alahmadi, M. Alkan and S.R. López-Permouth, Poor modules: The opposite of injectivity, Glas- gow Math. J. 52A, 7–17, 2010.
3. [3] U. Albrecht, J. Dauns and L. Fuchs, Torsion-freeness and non-singularity over right p.p.-rings, J. Algebra, 285 (1), 98–119, 2005.
4. [4] R. Alizade and Y. Durğun, Test modules for flatness, Rend. Semin. Mat. Univ. Padova, 137, 75–91, 2017.
5. [5] P. Aydoğdu and S. R. López-Permouth, An alternative perspective on injectivity of modules, J. Algebra, 338, 207–219, 2011.
6. [6] P. Aydoğdu and B. Saraç, On artinian rings with restricted class of injectivity domains, J. Algebra, 377, 49–65, 2013.
7. [7] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, C-pure projective modules, Comm. Algebra, 41 (12), 4559–4575, 2013.
8. [8] A.W. Chatters and C.R. Hajarnavis, Rings with chain conditions, Pitman, Boston, Mass.-London, 1980.

9. [9] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Birkhäuser Verlag, Basel, 2006.
10. [10] F. Couchot, RD-flatness and RD-injectivity, Comm. Algebra, 34 (10), 3675–3689, 2006.
11. [11] J. Dauns and L. Fuchs, Torsion-freeness for rings with zero divisor, J. Algebra Appl. 3 (3), 221–237, 2004.
12. [12] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Longman Scientific & Technical, 1994.
13. [13] Y. Durğun, Rings whose modules have maximal or minimal subprojectivity domain, J. Algebra Appl. 14 (6), 1550083, 2015.
14. [14] Y. Durğun, An alternative perspective on flatness of modules, J. Algebra Appl. 15 (8) 1650145, 2016.
15. [15] Y. Durğun, Subprojectivity domains of pure-projective modules, J. Algebra Appl. 19 (5), 2050091, 2020.
16. [16] E. E. Enochs and O. M. G. Jenda Relative homological algebra, Walter de Gruyter & Co., 2000.
17. [17] A. Facchini and A. Moradzadeh-Dehkordi, Rings over which every RD-projective module is a direct sums of cyclically presented modules, J. Algebra 401, 179–200, 2014.
18. [18] A. Hattori, A foundation of torsion theory for modules over general rings, Nagoya Math. J. 17, 147– 158, 1960.
19. [19] C. Holston, S.R. López-Permouth and N.O. Ertaş, Rings whose modules have maximal or minimal projectivity domain, J. Pure Appl. Algebra, 216 (3), 673–678, 2012.
20. [20] C. Holston, S.R. López-Permouth, J. Mastromatteo and J.E. Simental-Rodriguez, An alternative per- spective on projectivity of modules, Glasgow Math. J. 57 (1), 83–99, 2015.
21. [21] K. Honda, Realism in the theory of abelian groups. I, Comment. Math. Univ. St. Paul. 5, 37–75, 1956.
22. [22] T.Y. Lam, Lectures on modules and rings, Springer-Verlag, 1999.
23. [23] S.R. López-Permouth, J. Mastromatteo, Y. Tolooei and B. Ungor, Pure-injectivity from a different perspective, Glasg. Math. J. 60 (1), 135–151, 2018.
24. [24] L. Mao, Properties of RD-projective and RD-injective modules, Turkish J. Math. 35 (2), 187–205, 2011.
25. [25] L. Mao and N. Ding, On divisible and torsionfree modules, Comm. Algebra, 36 (2), 708–731, 2008.
26. [26] C. Megibben, Absolutely pure modules, Proc. Amer. Math. Soc. 26, 561–566, 1970.
27. [27] P. Rothmaler, Torsion-free, divisible, and Mittag-Leffler modules, Comm. Algebra 43 (8), 3342–3364, 2015.
28. [28] J. Rotman, An introduction to homological algebra, Academic Press, 1979.
29. [29] F. L. Sandomierski, Nonsingular rings, Proc. Amer. Math. Soc. 19, 225–230, 1968.
30. [30] J. Trlifaj, Whitehead test modules, Trans. Amer. Math. Soc. 348 (4), 1521–1554, 1996.
31. [31] R. B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28, 699–719, 1969.