A solution to the Baer splitting problem
Let R be a commutative domain. We prove that an R-module B is projective if and only if Ext^1(B,T)=0 for any torsion module T. This answers in the affirmative a question raised by Kaplansky in 1962.
math.AC↗
arXiv subjects
Dolors Herbera
Publications and source records attributed to Dolors Herbera.
Local morphisms and modules with a semilocal endomorphism ring
An associative ring with 1 is said to be semilocal provided it is semisimple artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it is isomorphic to a finite product of matrices over division rings. Modules with a semilocal endomorphism ring inherit some properties of semisimple modules. For example, they can be decomposed into a finite direct sum of indecomposable submodules, such decomposition is not unique but there are only a finite number of different ones, they cancel from direct sums and they satisfy the n-th root uniqueness property. A ring homomorphism is said to be local if it carries non-units to non-units. Semilocal rings can be characterized as those rings having a local homomorphism to a semisimple artinian ring. In this paper show that local homomorphisms appear frequently in module theory. As a consequence it will follow that many interesting classes of modules, as for example finitely presented modules over semilocal rings, have a semilocal endomorphism ring.
math.RA↗
Hamiltonian selfdistributive quasigroups
The problem of the existence of non-medial distributive hamiltonian quasigroups is solved. Translating this problem first to commutative Moufang loops with operators, then to ternary algebras and, finally, to cocyclic modules over\linebreak $\Bbb Z[x,x^{-1},(1-x)^{-1}]$, it is shown that every non-medial distributive hamiltonian quasigroup has at least 729 elements and that there are just two isomorphism classes of such quasigroups of the least cardinality. The quasigroups representing these two classes are anti-isomorphic.
math.RA↗