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

We prove that the ideal in complex cobordism ring ${\text{MU}}^{\ast }$ generated by the polynomial generators $S=(x_1,x_k,k\ge 3)$ of $c_1$-spherical cobordism ring $W^{\ast }$, viewed as elements in ${\text{MU}}^{\ast }$ by forgetful map is prime. Using the Baas-Sullivan theory of cobordism with singularities we define a commutative complex oriented cohomology theory ${\text{MU}}_S^{\ast }(-)$, complex cobordism modulo $c_1$-spherical cobordism, with the coefficient ring ${\text{MU}}^{\ast }\mathrm{/}S$. Then any $\mathrm{\Sigma }\subseteq S$ is also regular in ${\text{MU}}^{\ast }$ and therefore gives a multiplicative complex oriented cohomology theory ${\text{MU}}_{\mathrm{\Sigma }}^{\ast }(-)$. The generators of $W^{\ast }[1\mathrm{/}2]$ can be specified in such a way that for $\mathrm{\Sigma }=(x_k,k\ge 3)$ the corresponding cohomology is identical to the Abel cohomology, previously constructed in \cite{BUSATO}. Another example corresponding to $\mathrm{\Sigma }=(x_k,k\ge 5)$ gives the coefficient ring of the universal Buchstaber formal group law after tensored by $\mathbb{Z}[1\mathrm{/}2]$, i.e., is identical to the scalar ring of the Krichever-Hoehn complex elliptic genus \cite{KR}, \cite{H}.