From: wcw@math.psu.edu (William C Waterhouse)
Subject: Re: Direct sum of modules
Date: 25 Jan 2000 21:05:06 GMT
Newsgroups: sci.math
Summary: [missing]

In article <86begp$8fu$1@nnrp1.deja.com>, 
J. Agustín Peña A. <cuxito5@my-deja.com> writes:

>  Hi, guys! I have a problem here, I hope you can help me: Let A, B, C
> be three R-modules (R a ring with unity), such that A and B are
> submodules of C, and the intersection of A and B is the submodule 0.
> Let A(DS)B denote the direct sum of A and B. If A and B are direct
> summands of C, can we conclude A(DS)B is sirect summand of C too?


No, this doesn't follow.  For instance, take R to be the integers Z,
and let C be Z x Z. Let A be the submodule generated by (1,3), and
let B be the submodule generated by (1,5). These clearly have
intersection 0. If we let D be the submodule generated by (0,1), then
C is the direct sum of A and D and also the direct sum of B and D.
But all (m,n) in the direct sum of A and B have m+n even, so that
direct sum is not all of C (and cannot be a smaller direct summand, 
since the quotient is finite).

William C. Waterhouse
Penn State