Examples of modules as trivial extensions and algebraic topology type
by
Adnan Tercan
Hacettepe University Department of Mathematics

Let R be ring with identity and M an unital right R-module. In [3], (C₁₁)-modules worked out as a generalization of CS-modules. For a good account on CS-modules and concepts related to it refer to [1] and [2]. Recall that a module M is called (C₁₁)-module provided that every submodule of M has a complement which is a direct summand. Moreover a module M is called weak (C₁₁)-module if each of its semisimple submodules has a complement which is a direct summand. Clearly every (C₁₁)-module is weak (C₁₁) but the converse is not true in general. Note that any module with zero socle is a weak (C₁₁)-module. In this note firstly, we answer the following questions in the negative by constructing counter examples.
Question 1. Is a direct sum of a module with essential socle and a module with zero socle a weak (C₁₁)-module?
Question 2. Is a direct sum of a (C₁₁)-module with essential socle and a module with zero socle a (C₁₁)-module?
It was left open [3] whether direct summands of a (C₁₁)-module have (C₁₁) or not? Recently this question was answered in [4], namely; there exists a counter example which makes it clear that the (C₁₁) property is not inherited by direct summands unlike the CS property. Secondly, we obtain that not only one but also there are plentiful counter examples to the former question on direct summands of (C₁₁)-modules.

REFERENCES

[1] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Pitman Research Notes in Mathematics Series, Longman, Harlow, 1994.

[2] S.H. Mohammed and B.J. Muller, Continuous and discrete modules, London Math. Soc., Lecture Notes Series 147, Cambridge, 1990.

[3] P.F. Smith and A. Tercan, Generalizations of CS-modules, Comm. in Algebra, 21(6), 1809-1847 (1993).

[4] P.F. Smith and A. Tercan, Direct summands of modules with (C₁₁). Submitted.

Date received: June 1, 2000