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On Distinguishing Local Finite Rings from Finite Rings Only by Counting Elements and Zero Divisors

Journal Name:

  • European Journal of Pure and Applied Mathematics

Publication Year:

  • 2014

Key Words:

Author NameUniversity of Author
Abstract (2. Language): 
The purpose of this short communication is to prove the following: Let R be a finite associative ring with unit. Then R is local if and only if |R| = pn and |Z(R)| = pm for some prime number p and integers 1 < m < n. For the commutative case, this have been recently discovered by Behboodi and Beyranvand [1, Theorem 3]. We will also present yet another proof for the commutative case.
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REFERENCES

References: 

[1] M. Behboodi and R. Beyranvand. On the Structure of Commutative Rings with pk • • • pj^ (1 < k < 7) zero divisors. European Journal of Pure and Applied Mathematics, 3(2):303-316, 2010.
[2] N. Ganesan. Properties of rings with a finite number of zero-divisors II. Mathematische Annale, 161:241-246, 1965.
[ 3] R. Gilmer. Zero-divisors in commutative rings. The American Mathematical Monthly, 93(5):382-387, 1986.
[4] B. R. McDonald. Finite Rings with Identity. Marcel Dekker, New York, 1974.
[5] R. Raghavendran. Finite associative rings. Compositio Mathematica, 21:195-229, 1969.
[6] H. Wielandt. Ein Beweis für die Existenz der Sylowgruppen. Archiv der Mathematik, 10(1):401-402, 1959.

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