Geodesic
$RD$-formulas and $W$-rings
Matematičeskie zametki, Tome 53 (1993) no. 1, pp. 95-103.

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We study rings over which any finitely presented module is a direct summand of a direct sum of cyclic cyclically presented modules (Warfield rings). We characterize these rings in matrix terms. We clarify a number of properties of Warfield rings including their complete characterization in some special cases. We prove the presence of special (deeper than $pp$) elimination of quantifiers in module theory over these rings. 
@article{MZM_1993_53_1_a9,
     author = {G. E. Puninskii},
     title = {$RD$-formulas and $W$-rings},
     journal = {Matemati\v{c}eskie zametki},
     pages = {95--103},
     publisher = {mathdoc},
     volume = {53},
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     year = {1993},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1993_53_1_a9/}
}
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G. E. Puninskii. $RD$-formulas and $W$-rings. Matematičeskie zametki, Tome 53 (1993) no. 1, pp. 95-103. http://geodesic.mathdoc.fr/item/MZM_1993_53_1_a9/