Homological aspects of formal triangular matrix rings

Anahtar Kelimeler:

formal triangular matrix ring, projective dimension, injective dimension, flat dimension, FP-injective dimension

Homological aspects of formal triangular matrix rings

Let $T=\biggl(\begin{matrix} A&0\\U&B\end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We first give some computing formulas of projective, injective, flat and $FP$-injective dimensions of special left $T$-modules. Then we establish some formulas of (weak) global dimensions of $T$. It is proven that (1) If $U_{A}$ is flat and $_{B}U$ is projective, $lD(A)\neq lD(B)$, then $lD(T)={\rm max}\{lD(A),lD(B)\}$; (2) If $U_{A}$ and $_{B}U$ are flat, $wD(A)\neq wD(B)$, then $wD(T)={\rm max}\{wD(A),wD(B)\}$.

Keywords:

formal triangular matrix ring, projective dimension, injective dimension, flat dimension, FP-injective dimension,

PDF

___

[1] J. Asadollahi and S. Salarian, On the vanishing of Ext over formal triangular matrix rings, Forum Math. 18, 951-966, 2006.
[2] R.R. Colby, Rings which have flat injective modules, J. Algebra 35, 239-252, 1975.
[3] R.R. Colby and K.R. Fuller, Equivalence and Duality for Module Categories, Cambridge University Press, Cambridge, 2004.
[4] N.Q. Ding and J.L. Chen, The flat dimensions of injective modules, Manuscripta Math. 78, 165-177, 1993.
[5] D.J. Fieldhouse, Character modules, dimension and purity, Glasgow Math. J. 13, 144-146, 1972.
[6] R.M. Fossum, P. Griffith and I. Reiten, Trivial Extensions of Abelian Categories, Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory. Lect. Notes in Math. 456, Springer-Verlag, Berlin, 1975.
[7] K.R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Monographs Textbooks Pure Appl. Math. 33, Marcel Dekker, Inc. New York and Basel, 1976.
[8] R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, GEM 41, De Gruyter, Berlin-New York, 2006.
[9] E.L. Green, On the representation theory of rings in matrix form, Pacific J. Math. 100, 123-138, 1982.
[10] A. Haghany and K. Varadarajan, Study of formal triangular matrix rings, Comm. Algebra 27, 5507-5525, 1999.
[11] A. Haghany and K. Varadarajan, Study of modules over formal triangular matrix rings, J. Pure Appl. Algebra 147, 41-58, 2000.
[12] P. Krylov and A. Tuganbaev, Formal Matrices, Springer International Publishing, Switzerland, 2017.
[13] T.Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York-Heidelberg- Berlin, 1999.
[14] P. Loustaunau and J. Shapiro, Homological dimensions in a Morita context with applications to subidealizers and fixed rings, Proc. Amer. Math. Soc. 110, 601-610, 1990.
[15] L.X. Mao, Cotorsion pairs and approximation classes over formal triangular matrix rings, J. Pure Appl. Algebra 224, 106271 (21 pages), 2020.
[16] L.X. Mao, Duality pairs and FP-injective modules over formal triangular matrix rings, Comm. Algebra 48, 5296-5310, 2020.
[17] L.X. Mao, The structures of dual modules over formal triangular matrix rings, Publ. Math. Debrecen 97 (3-4), 367-380, 2020.
[18] L.X. Mao, Homological dimensions of special modules over formal triangular matrix rings, J. Algebra Appl. 21, 2250146 (14 pages), 2022.
[19] C. Psaroudakis, Homological theory of recollements of abelian categories, J. Algebra 398, 63-110, 2014.
[20] J.J. Rotman, An Introduction to Homological Algebra, Second Edition, Springer, New York, 2009.
[21] B. Stenström, Coherent rings and FP-injective modules, J. London Math. Soc. 2, 323-329, 1970.