We prove that a sequence $S=(x_1, x_k, k\geq 3)$ of polynomial generators of
$c_1$-spherical cobordism ring $W_*$, viewed as a sequance in the complex
cobordism ring \MU_* by forgetful map, is regular. Using the Baas-Sullivan
theory of cobordism with singularities we define a commutative complex oriented
cohomology theory \MU^*_S(-), complex cobordism modulo $c_1$-spherical
cobordism, with the coefficient ring \MU_*/S. Then any $\Sigma\subseteq S$ is
also regular in \MU^* and therefore gives a multiplicative complex oriented
cohomology theory \MU^*_{\Sigma}(-). The generators of $W_*$ can be specified
in such a way that for $\Sigma=(x_k, k\geq 3)$ the corresponding cohomology is
identical to the Abel cohomology, previously constructed in \cite{BUSATO}.
Another example corresponding to $\Sigma=(x_k, k\geq 5)$ is classified by the
Krichever-Hoehn complex elliptic genus \cite{KR}, \cite{H} modulo torsion.Comment: 10 page