id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-1802	González, Marcos José	On Distinguishing Local Finite Rings from Finite Rings Only by Counting Elements and Zero Divisors	2014	5	.pdf	application/pdf	2024	131	84	Then (i) Z(R) is the Jacobson radical of R; (ii) |R|= pnr, and |J|= p(n−1)r for some prime number p, and some positive integers n,r; (iii) Z(R)n New Proof of Theorem 2 In this section, we shall provide a new proof for Theorem 2 based on the fact that, for the case in which R is a finite commutative ring, the Jacobson radical J(R) coincides with the ideal of all the nilpotent elements of the ring R. Let R be finite commutative ring with identity of order |R|= pm and having |Z(R)|	cache/ejpam-1802.pdf	txt/ejpam-1802.txt