Geodesic
A Simple Proof of a Theorem on Reduced Rings
Canadian mathematical bulletin, Tome 23 (1980) no. 4, pp. 495-496

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We give a simple proof of a theorem by Andrunakievič and Rjabuhin which states that a reduced ring is a subdirect product of entire rings. Our proof makes no use of m-systems and is in some sense similar to the proof of the corresponding theorem in the commutative case due to Krull.A reduced ring is a ring without non-zero nilpotent elements. It is wellknown that if a reduced ring is commutative, then it is a subdirect product of integral domains [2]. This result has been generalized to arbitrary reduced rings [1]. The proof in the general case is somewhat complicated. We present a simple proposition which leads to a simple proof of the general case. 
Klein, Abraham A. A Simple Proof of a Theorem on Reduced Rings. Canadian mathematical bulletin, Tome 23 (1980) no. 4, pp. 495-496. doi: 10.4153/CMB-1980-075-7
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     author = {Klein, Abraham A.},
     title = {A {Simple} {Proof} of a {Theorem} on {Reduced} {Rings}},
     journal = {Canadian mathematical bulletin},
     pages = {495--496},
     year = {1980},
     volume = {23},
     number = {4},
     doi = {10.4153/CMB-1980-075-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1980-075-7/}
}
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