Narrow Results

We describe commutative rings of prime characteristic with the property indicated in the title of the paper.

We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be reduced to the study of proper polynomial identities. In particular, the factors of series of ℤS_n-submodules in the ℤS_n-modules of multilinear polynomial functions can be derived by the analog of Young's (or Pieri's) rule from the factors of series in the corresponding ℤS_n-modules of proper polynomial functions. As an application, we calculate the codimensions and a basis of multilinear polynomial identities of unitary rings of upper triangular 2 × 2 matrices and infinitely generated Grassmann algebras over unitary rings. In addition, we calculate the factors of series of ℤS_n-submodules for these algebras. Also we establish relations between codimensions of rings and codimensions of algebras and show that the analog of Amitsur's conjecture holds in all torsion-free rings, and all torsion-free rings with 1 satisfy the analog of Regev's conjecture.

The paper deals with new specific constructions of indecomposable torsion-free abelian groups of rank two and nonzero rings on them. They illustrate purely theoretical results and complement quite rare examples obtained during the classical as well as recent research of additive groups of rings. The presented results concerning the homogeneous groups remain true for groups of any finite nonzero rank. Moreover, the paper contains a construction of a torsion-free indecomposable abelian group of an arbitrary finite rank greater than two supporting an associative, but not commutative ring, as well as a ring which is neither associative nor commutative.

An abelian group $A$ is called a $TI$-group if every associative ring with the additive group $A$ is filial. The filiality of a ring $R$ means that the ring $R$ is associative and all ideals of any ideal of $R$ are ideals in $R$. In this paper, torsion-free $TI$-groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition $TI$ implies the indecomposability and homogeneity. The paper contains constructions of $2^{{\aleph }_0}$ such groups of any rank from 2 to$2^{{\aleph }_0}$ which are pairwise non-isomorphic.