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We prove that the ideal in complex cobordism ring $\MU^*$ generated by thepolynomial generators $S=(x_1, x_k, k\geq 3)$ of $c_1$-spherical cobordism ring$W^*$, viewed as elements in $\MU^*$ by forgetful map is prime. Using theBaas-Sullivan theory of cobordism with singularities we define a commutativecomplex 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 amultiplicative complex oriented cohomology theory $\MU^*_{\Sigma}(-)$. Thegenerators of $W^*[1/2]$ 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$\Sigma=(x_k, k\geq 5)$ gives the coefficient ring of the universal Buchstaberformal group law after tensored by $\mathbb{Z}[1/2]$, i.e., is identical to thescalar ring of the Krichever-Hoehn complex elliptic genus \cite{KR}, \cite{H}.

Complex cobordism modulo $c_1$-spherical cobordism and related genera