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ABSTRACT. We show that each of the irreducible components of moduli of rank 2 torsionfree sheaves with odd Euler characteristic over a reducible nodal curve is rational.

We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.

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A ring [Formula: see text] is said to be i-reversible if for every [Formula: see text] [Formula: see text][Formula: see text], [Formula: see text] is a nonzero idempotent implies [Formula: see text] is an idempotent. It is known that the rings [Formula: see text] and [Formula: see text] (the ring of all upper triangular matrices over [Formula: see text]) are not i-reversible for [Formula: see text]. In this paper, we provide a nontrivial i-reversible subring of [Formula: see text] when [Formula: see text] and [Formula: see text] has only trivial idempotents. We further provide a maximal i-reversible subring of [Formula: see text] for each [Formula: see text], if [Formula: see text] is a field. We then give conditions for i-reversibility of trivial, Dorroh and Nagata extensions. Finally, we give some independent sufficient conditions for i-reversibility of polynomial rings, and more generally, of skew polynomial rings.