TY - GEN
T1 - A note on (α, β)-higher derivations and their extensions to modules of quotients
AU - Vaš, Lia
AU - Papachristou, Charalampos
N1 - Publisher Copyright:
© 2010 Springer Basel AG.
PY - 2010
Y1 - 2010
N2 - We extend some recent results on the differentiability of torsion theories. In particular, we generalize the concept of (α, β)-derivation to (α, β)-higher derivation and demonstrate that a filter of a hereditary torsion theory that is invariant for α and β is (α, β)-higher derivation invariant. As a consequence, any higher derivation can be extended from a module to its module of quotients. Then, we show that any higher derivation extended to a module of quotients extends also to a module of quotients with respect to a larger torsion theory in such a way that these extensions agree. We also demonstrate these results hold for symmetric filters as well. We finish the paper with answers to two questions posed in [L. Vaš, Extending higher derivations to rings and modules of quotients, International Journal of Algebra, 2 (15) (2008), 711-731]. In particular, we present an example of a non-hereditary torsion theory that is not differential.
AB - We extend some recent results on the differentiability of torsion theories. In particular, we generalize the concept of (α, β)-derivation to (α, β)-higher derivation and demonstrate that a filter of a hereditary torsion theory that is invariant for α and β is (α, β)-higher derivation invariant. As a consequence, any higher derivation can be extended from a module to its module of quotients. Then, we show that any higher derivation extended to a module of quotients extends also to a module of quotients with respect to a larger torsion theory in such a way that these extensions agree. We also demonstrate these results hold for symmetric filters as well. We finish the paper with answers to two questions posed in [L. Vaš, Extending higher derivations to rings and modules of quotients, International Journal of Algebra, 2 (15) (2008), 711-731]. In particular, we present an example of a non-hereditary torsion theory that is not differential.
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U2 - 10.1007/978-3-0346-0007-1_12
DO - 10.1007/978-3-0346-0007-1_12
M3 - Conference contribution
AN - SCOPUS:84959122977
SN - 9783034600064
T3 - Trends in Mathematics
SP - 165
EP - 174
BT - Ring and Module Theory
A2 - Albu, Toma
A2 - Birkenmeier, Gary F.
A2 - Erdoǧan, Ali
A2 - Tercan, Adnan
PB - Springer International Publishing
T2 - International Conference on Ring and Module Theory, 2008
Y2 - 18 August 2008 through 22 August 2008
ER -