My primary interest is in rings, ideals and modules .  I work with projective modules over commutative, Noetherian rings. A projective module is a direct summand of a free module. Projective modules can also be thought of as algebraic analogue of vector bundles. One example of a projective module which is not free is the tangent bundle of the two-dimensional real sphere (proof of which follows from the so called "Hairy ball theorem"). One central question is to find necessary and sufficient conditions for a projective module to split off a free summand of rank one (or, a vector bundle to split off a trivial bundle). In a situation when rank of the projective module is equal to the Krull dimension of the ring (on which the module is defined), there are some invariants which take care of the above question. For instance, if the ring is the coordinate ring of an affine variety (smooth) over an algebraically closed field, the Chern class of the projective module is the desired invariant, which vanishes in the Chow group if and only if the projective module splits off a free, rank one summand. But example of tangent bundle mentioned above shows that the Chern class may vanish without the projective module splitting a free summand of rank one, thus implying the requirement of a better invariant than Chern class in cases when the projective module is defined on an affine domain over a field which is not algebraically closed (for example, affine varieties over real numbers). Motivation and model of this invariant comes from topology. It is the Euler class. At present I work in this Euler class theory.