Flat and FP-injectivity

Abstract

One of the main results of this paper is the characterization of left FP-injective rings as rings over which every finitely presented right module is torsionless. The necessity of the torsionless condition (in fact reflexivity) previously has been observed for special classes of FP-injective rings (e.g., self-injective rings (Kato and McRae), right and left coherent and self FP-injective rings (Stenström)). Furthermore, if the ring is left coherent, it is equivalent to say that every finitely presented right module is reflexive. We investigate the condition (called IF) that every injective right module is flat and characterize this via FP-injectivity. IF-rings generalize regular rings and QF-rings, and under certain chain or homological conditions coincide with regular or QF-rings (e.g., <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w g l dimension upper R less-than-over-equals 1 right double arrow regular"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mi>g</mml:mi> <mml:mi>l</mml:mi> <mml:mi>dim</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>R</mml:mi> <mml:mo> ≦ </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false"> ⇒ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>regular</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">wgl\dim R \leqq 1 \Rightarrow {\text {regular}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , noetherian, or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="right perfect right double arrow QF right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>right perfect</mml:mtext> </mml:mrow> <mml:mo stretchy="false"> ⇒ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>QF)</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {right perfect}} \Rightarrow {\text {QF)}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .