## Abstract

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 $2\times 2$ 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.

As an application, we calculate the codimensions and a basis of multilinear polynomial identities of unitary rings of upper triangular $2\times 2$ 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.

Original language | English |
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Pages (from-to) | 1925-1943 |

Number of pages | 19 |

Journal | International Journal of Algebra and Computation |

Volume | 23 |

Issue number | 08 |

Publication status | Published - Dec 2013 |

## Keywords

- polynomial identity
- integral representation
- group ring
- codimension