On the indecomposability of induced modules

Reinhard Knörr
1986 Journal of Algebra  
Then v is indecomposable for every indecomposable direct summand W of SH. Moreover, if W, and W2 are two surh .summands, then WY z WF if and only 17 W, E W,. Remarks. (I ) If R = K, then the result is an easy consequence of Frobenius reciprocity; the proof given below, however, covers also this case. (2) A well known and much used special instance arises in case U = N. Condition (i) is then just that S is indecomposable, while (ii) asserts that HZ TJS). If R is a field, the theorem is therefore
more » ... precisely Theorem 9.6(a), (c) in [3, Chap. VII]. Note that part (b) of this theorem does not hold for UC-lattices, while parts (a) and (c) do; in fact, their proofs need only the Krull-Schmidt theorem and rank arguments, so carry over directly. We will use this special case in proving the more general
doi:10.1016/0021-8693(86)90214-0 fatcat:copk6wgavzcodmeysgb64kf7qm