Isomorphism between direct sum of modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:03:48Z http://mathoverflow.net/feeds/question/24697 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24697/isomorphism-between-direct-sum-of-modules Isomorphism between direct sum of modules Golden Field 2010-05-15T05:11:08Z 2010-06-30T00:02:50Z <p>Let \$M\$, \$N\$ be two modules over ring \$A\$. If \$M\oplus M\cong N\oplus N\$, can we conclude \$M\cong N\$? In the case that \$M\$, \$N\$ are completely decomposable (e.g. finite-length module by Krull-Schmidt Theorem), it is easy to show this must be true. Does the general case also hold?</p> http://mathoverflow.net/questions/24697/isomorphism-between-direct-sum-of-modules/29986#29986 Answer by Greg Marks for Isomorphism between direct sum of modules Greg Marks 2010-06-30T00:02:50Z 2010-06-30T00:02:50Z <p>There are even counterexamples in the case \$A = {\mathbb Z}\$: at the end of B. J&#x00F3;nsson&#8217;s paper &#8220;On direct decompositions of torsion-free abelian groups,&#8221; <i> Math. Scand.</i> <b>5</b> (1957), 230&#8211;235, an example is given of torsion-free, finite-rank abelian groups \$B \not\cong C\$ such that \$B \oplus B \cong C \oplus C\$.</p> <p>A further counterexample, which I believe has been pointed out independently by L. S. Levy, R. Wiegand, and R. G. Swan: let \$A\$ be the coordinate ring of the real 2-sphere and \${}_AM\$ the module for the tangent bundle; then \$M \oplus M\$ is free of rank \$4,\$ but \$M\$ is not free of rank \$2\$.</p> <p>In the positive direction, K. R. Goodearl has proved (&#8220;Direct sum properties of quasi-injective modules,&#8221; <i>Bull. Amer. Math. Soc.</i> <b>82</b> (1976), no. 1, 108&#8211;110, Theorem 3) that if \$M\$ and \$N\$ are <i>quasi-injective</i> modules over a ring (commutative or not), then \$M^n \cong N^n\$ implies \$M \cong N\$ for any positive integer \$n\$.</p> <p>Your question is related to an important open problem in noncommutative ring theory, the &#8220;separativity&#8221; problem for von Neumann regular rings: if \$R\$ is a von Neumann regular ring (or more generally an exchange ring), and \$A\$ and \$B\$ are finitely generated projective left \$R\$-modules with the property that \$A \oplus A \cong A \oplus B \cong B \oplus B\$, must we have \$A \cong B\$?&#160; An affirmative answer would resolve several major open problems, as explained in P. Ara, K. R. Goodearl, K. C. O&#8217;Meara, and E. Pardo&#8217;s paper &#8220;Separative cancellation for projective modules over exchange rings,&#8221; <i>Israel J. Math.</i> <b>105</b> (1998), 105&#8211;137. </p>