QUASI-PROJECTIVE MODULES, PERFECT MODULES, AND A THEOREM FOR MODULAR LATTICES
Yôichi Miyashita · Hokkaido Mathematical Journal · 1966
\S 0 Introduction. .86 \S 1 Preliminary results and perfect modules.87 \S 2 Quasi-projective modules.91 \S 3 Perfect projective modules.96 \S 4 Semi-perfect h-central rings.101 \S 5 A theorem for modular lattices.103 \S 6 Lemmas on radical and perfectness. 108\S 0. Introduction.Throughout the present paper, $R$ is a ring with 1, and $M$ a unital R-left module."Submodule" and "homomorphism" will mean "R-submodule" and "R-homomorphism", respectively.For any R-submodule $A$ of $M$ , we denote by $ u(M\rightarrow M/A)$ and $c(A\rightarrow M)$ the projection of $M$ onto $M/A$ and the injection of $A$ into $M$ , respectively.$M$ is called R-perfect if for any pair of submodules $A,$ $B$ of $M$ with $A+B=M$ there exists a submodulefor all proper submodule $X$ .Then, $\Re(RM)$ (radical of $M,$ $i.e$ .the meet of all maximal submodules of $M$ ) $=the$ sum of all d-dense submodules (Prop. 1.4).From this point of view, we should like to have another look at radicals of modules (Th.2.11, Th. 2.12 and Th.4.3).projective and $A,$ $B$ are d-complements of each other then $M=A\oplus B$ .By the light of this fact, we shall prove the following: If ${}_{R}P$ is an R-projective module, then the following conditions for $P$ are equivalent.(i) $P$ is R-perfect.(ii) Every homomorphic image of $P$ has a projective cover.(iii) $\Re(RP)$ is d-dense in $P$ , and $P$ is a direct sum of sum-irreducible submodules, where a sum-irreducible module is a module such that the sum of its two proper submodules is always proper (Th.3.3 and Th. 3.7).$R$ is called a semi-perfect ring, if $RR$ is perfect, andProof.Let $B$ be a d-complement of $A$ .If $(A\cap B)+X=B$ for some submodule $X$ of $B$ , then $M=A+B=A+X$ .The minimality of $B$ implies $X=B$.$HenceA\cap Bisd-denseinB$ .Conversely, $assumethatA\cap Bisd$ -dense in $B$ .Let $A+B_{0}=M$ for some submodule $B_{0}$ of $B$ .Then $B=B\cap(A+B_{0})$ $=(A\cap B)+B_{0}$ .Since $A\cap B$ is d-dense in $B$ , we have $B=B_{0}$ .Hence $B$ is a d-complement of $A$ .Let $D$ be a d-dense submodule of $M$ , and let $B=(D\cap B)$ $+X$ for some submodule $X$ .Then $M=A+B=A+(D\cap B)+X$ Since $D\cap B(\subseteqq D)$ is d-dense in $M$ , we have $M=A+X$.Then, the minimality of $B$ implies $X=B$.Hence $D\cap B$ is d-dense in $B$ .By Prop. 1.2, $\Re(RB)\subset=$ $\Re(RM)\cap B$ .Conversely, for any $x\in\Re(RM)\cap B,$ $Rx$ is d-dense in $M$ , and therefore $Rx=Rx\cap B$ is d-dense in $B$ .