Let Mnbe a closed smooth manifold andT:Z₂×Mⁿ→Mⁿ denote a smooth action of the integral additive groupZ₂ on Mnwhich is called involution . The fixed point set F of T is the disjoint union of closed manifold Mn ,which are finite in number . If each componentof F isof constant dimension n -k ,we say that F is of constant codimension k . Let R n be the group of unoriented bordism class of n -dimensional smooth manifolds and let Jkn be its subset consisting of the classes which are represented by manifolds admitting smooth involutions with fixed point set of constant codimension k . It is easy to see that Jkn is a subgroup of R n and thatJ^k_*=∑^∞_n=kJ^k_* is an ideal of the unoriented bordism ring R_*=∑R_{n<>/sub. The case k =10 by means of constructing the generators ofR* is discussed.}