Common use of Theorem 5 Clause in Contracts

Theorem 5. There exists a deterministic polynomial-time algorithm that, given a finite ring R and a R-module M, together with a generating set of cardinality d, for some d Z 0, determines if M is R-projective or not, and if it is, produces a splitting of the natural surjection Rd → M. Proof. Recall that M is projective if and only if the natural surjection f : Rd → M has a left inverse. The latter can be tested using Proposition 2.5.1, which will also produce a left inverse. 68 Algorithms for finite rings Second proof. Another way to determine whether M is projective comes as a conse- quence of Theorem 4.1.1, since M is projective if and only if M is a direct summand of Rd. We compute the largest isomorphic common direct summand of Rd and M , say S. If M =~ S, then M is projective and the isomorphism M S, which is also produced by the algorithm, induces a splitting of ▇▇ ▇ . Otherwise the algorithms concludes that M is not projective.