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On the structure of sequentially Cohen-Macaulay bigraded modules

Leila Parsaei Majd, Ahad Rahimi (2015)

Czechoslovak Mathematical Journal

Let K be a field and S = K [ x 1 , ... , x m , y 1 , ... , y n ] be the standard bigraded polynomial ring over K . In this paper, we explicitly describe the structure of finitely generated bigraded “sequentially Cohen-Macaulay” S -modules with respect to Q = ( y 1 , ... , y n ) . Next, we give a characterization of sequentially Cohen-Macaulay modules with respect to Q in terms of local cohomology modules. Cohen-Macaulay modules that are sequentially Cohen-Macaulay with respect to Q are considered.

On ( σ , τ ) -derivations in prime rings

Mohammad Ashraf, Nadeem-ur-Rehman (2002)

Archivum Mathematicum

Let R be a 2-torsion free prime ring and let σ , τ be automorphisms of R . For any x , y ∈ R , set [ x , y ] σ , τ = x σ ( y ) - τ ( y ) x . Suppose that d is a ( σ , τ ) -derivation defined on R . In the present paper it is shown that ( i ) if R satisfies [ d ( x ) , x ] σ , τ = 0 , then either d = 0 or R is commutative ( i i ) if I is a nonzero ideal of R such that [ d ( x ) , d ( y ) ] = 0 , for all x , y ∈ I , and d commutes with both σ and τ , then either d = 0 or R is commutative. ( i i i ) if I is a nonzero ideal of R such that d ( x y ) = d ( y x ) , for all x , y ∈ I , and d commutes with τ , then R is commutative. Finally a related result has been obtain for ( σ , τ ) -derivation....

Orderings and preorderings in rings with involution

Ismail Idris (2000)

Colloquium Mathematicae

The notions of a preordering and an ordering of a ring R with involution are investigated. An algebraic condition for the existence of an ordering of R is given. Also, a condition for enlarging an ordering of R to an overring is given. As for the case of a field, any preordering of R can be extended to some ordering. Finally, we investigate the class of archimedean ordered rings with involution.

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