Geodesic
Rings with No Nilpotent Elements and with the Maximum Condition on Annihilators
Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 35-38

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Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3. 
Cornish, W. H.; Stewart, P. N. Rings with No Nilpotent Elements and with the Maximum Condition on Annihilators. Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 35-38. doi: 10.4153/CMB-1974-006-1
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     title = {Rings with {No} {Nilpotent} {Elements} and with the {Maximum} {Condition} on {Annihilators}},
     journal = {Canadian mathematical bulletin},
     pages = {35--38},
     year = {1974},
     volume = {17},
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     doi = {10.4153/CMB-1974-006-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-006-1/}
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