E-RINGS OF LOW RANKS
https://doi.org/10.22405/2226-8383-2017-18-2-235-244
Abstract
An associative ring R is called an E-ring if all endomorphisms of its additive group R+ are left multiplications, that is, for any α ∈ EndR+ there is r ∈ R such that α(x) = x · r for all x ∈ R. E-rings were introduced in 1973 by P. Schultz. A lot of articles are devoted to E-rings. But most of them are considered torsion free E-rings. In this work we consider E-rings (including mixed rings) whose ranks do not exceed 2. It is well known that an E-ring of rank 0 is exactly a ring classes of residues. It is proved that E-rings of rank 1 coincide with infinite T-ring (with rings Rχ). The main result of the paper is the description of E-rings of rank 2. Namely, it is proved that an E-ring R of rank 2 or decomposes into a direct sum of E-rings of rank 1, or R = Zm ⊕ J, where J is an m-divisible torsion free E-ring, or ring R is S-pure embedded in the ring ∏︀tp(R). In addition, we obtain some results about nilradical of a mixed
p∈S E-ring.
About the Author
A. V. Tsarev
Moscow Pedagogical State University
Russian Federation
Doctor of Physical and Mathematical Sciences, Professor of the Department of Algebra
Russian Federation
Doctor of Physical and Mathematical Sciences, Professor of the Department of Algebra
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Review
For citations:
Tsarev A.V. E-RINGS OF LOW RANKS. Chebyshevskii Sbornik. 2017;18(2):235-244. (In Russ.) https://doi.org/10.22405/2226-8383-2017-18-2-235-244
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