Failure of the Krull-Schmidt theorem for artinian modules and serial modules

The purpose in writing this note has been three-fold. First, we wanted to present the solution of a problem posed by Wolfgang Krull in 1932 [K]. Krull asked whether what is now called the "Krull-Schmidt Theorem" holds for artinian modules. A negative answer was published only in 1995 by Herbera, Levy, Vamos and the author [FHLV]. Second, we wanted to present the answer to a question posed by Warfield in 1975 [W2], namely, whether the Krull-Schmidt Theorem holds for serial modules. The author published a negative answer in 1996 [F1]. The solution to Warfield's problem shows an interesting behavior. Briefly, the Krull-Schmidt Theorem holds for some classes of modules and not for others. When it does hold, any indecomposable decomposition is uniquely determined up to a permutation. For serial modules the Krull-Schmidt Theorem does not hold, but any indecomposable decomposition is uniquely determined up to two permutations. Third, we wanted to present the structure of the semigroup S+(P-Mod R) of isomorphism classes of finitely generated projective modules over a semilocal ring R [FH1]. Both artinian modules and serial modules of finite Goldie dimension have semilocal endomorphism ring. Complete proofs can be found in part in the monograph [F2] and in part in the paper [FH1]. We shall consider unital right modules over an associative ring R with identity 1(R) not equal 0(R). If k is a commutative ring, a module-finite k-algebra is a k-algebra R which is finitely generated as a k-module. For any ring R we denote the Jacobson radical of R by J(R).