Forcing linearity numbers for coatomic modules

Year 2018, Volume: 1 Issue: 1, 1 - 4, 30.09.2018

Peter R. FUCHS

https://doi.org/10.33434/cams.446020

Abstract

We show that an integer $ n\in \mathbb{N}\cup \lbrace 0 \rbrace $ is the forcing linearity number of a coatomic module over an arbitrary commutative ring with identity if and only if $n\in \left\{ 0,1,2,\infty \right\} \cup \left\{ q+2\left\vert q\text{ is a prime power}\right. \right\} .$

Keywords

Homogeneous functions, Forcing linearity numbers, Coatomic modules

References

• [1] C.Faith, Algebra. II. Ring theory. Grundlehren der Mathematischen Wissenschaften, No. 191. Springer- Verlag, Berlin- New York, 1976.
• [2] R.M.Hamsher, Commutative rings over which every module has a maximal submodule, Proc. Amer. Math. Soc. 18 (1967), 1133- 1137.
• [3] C.J.Maxson, J.H.Meyer, Forcing linearity numbers, J.Algebra 223 (2000), 190- 207.
• [4] C.J.Maxson, A.B.Van der Merwe, Forcing linearity numbers for modules over rings with nontrivial idempotents, J.Algebra 256 (2002), 66- 84.
• [5] C.J.Maxson, A.B.Van der Merwe, Forcing linearity numbers for finitely generated modules, Rocky Mountain J.Math. 35 (3) (2005), 929-939.
• [6] A.A.Tuganbaev, Rings whose nonzero modules have maximal submodules, J.Math.Sci. (New York) 109 (2002), no.3, 1589- 1640.
• [7] H. Zöschinger, Koatomare Moduln, Math. Z. 170 (1980), 221- 232.