On finitely generated modules whose first nonzero Fitting ideals are regular

Document Type: Research Paper


Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.


A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of  $R^n$ which is generated by columns of  a matrix $A=(a_{ij})$ with $a_{ij}\in R$ for all $1\leq i\leq n$, $j\in \Lambda$, where $\Lambda $ is a (possibly infinite) index set.  Let  $M=R^n/N$ be  a module of type ($F_{n-1}$) and ${\rm T}(M)$ be the submodule of $M$ consisting of all elements of $M$ that are annihilated by a regular element of $R$. For $ \lambda\in \Lambda $, put $M_\lambda=R^n/<(a_{1\lambda},...,a_{n\lambda})^t>$. The main result of this paper asserts that if $M_\lambda $ is a regular $R$-module, for some $\lambda\in\Lambda$, then $M/{\rm T}(M)\cong M_\lambda/{\rm T}(M_\lambda)$. Also it is shown that if $M_\lambda$ is a regular torsionfree $R$-module, for some $\lambda\in \Lambda$, then $ M\cong M_\lambda. $ As a consequence we characterize all  non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.


[1] Brown, W.C., ``Matrices Over Commutative Rings", Pure Appl. Math. 169, Marcel Dekker Inc., 1993.
[2] Buchsbaum, D.A. and Eisenbud, D., What makes a complex exact?, J. Algebra 25 (1973), 259-268.
[3] Eisenbud, D., ``Commutative Algebra with a View toward Algebraic Geometry", Springer-verlag, 1995.
[4] Fitting, H., ``Die Determinantenideale eines Moduls", Jahresbericht d. Deutschen Math.-Vereinigung, 46 (1936), 195-228.
[5] Gopalakrishnan, N.S., ``Commutative Algebra", Oxonian press New Delhi, 1984.
[6] Lipman, J., On the Jacobian ideal of the module of di erentials, Proc. Amer. Math. Soc. 21 (1969), 423-426.
[7] Ohm, J., On the first nonzero Fitting ideal of a module, J. Algebra 320 (2008), 417-425.