schmidt theorem मीनिंग इन हिंदी

उदाहरण वाक्य

1. The following weak form of the Krull-Schmidt theorem holds.
2. This is part of the Krull Schmidt theorem, and holds more generally for finite direct products.
3. The Krull Schmidt theorem and related results give conditions under which a module has something like a primary decomposition, a decomposition as a direct sum of indecomposable modules in which the summands are unique up to order.
4. The weak form of the Krull-Schmidt Theorem holds not only for uniserial modules, but also for several other classes of modules ( biuniform modules, cyclically presented modules over serial rings, kernels of morphisms between indecomposable injective modules, couniformly presented modules .)
5. The main reason for this is that the structure of the collection of indecomposables is extremely complicated by Gabriel's theorem in the theory of quiver representations, although a finitely n-dim persistence module can be uniquely decomposed into a direct sum of indecomposables due to the Kull-Schmidt theorem.
6. The present-day Krull Schmidt theorem was first proved by Joseph Wedderburn ( " Ann . of Math " ( 1909 ) ), for finite groups, though he mentions some credit is due to an earlier study of G . A . Miller where direct products of abelian groups were considered.
7. In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull-Schmidt theorem : every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique ( meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic ).