Remarks on the Quasi-Frojectiveness of Ring

Abstract

R을 unity 1≠0을 가진 環이라하고 M을 unital 右 R-module이라 하자. M이 準射影的이라 함은 N을 M의 한 submodule이라 할때 任意의 準同型 寫像 f:M->M/N에 대하여 어떤 endomorphism f:M->M가 있어서 ν:M->M/N을 自然的 準同形對應이라 하면 f=??가 成立함을 말한다.

우리는 여기서 環 R을 하나의 R-mdule로 보았을때 R의 모든 準同型像이 準射影的일 必要條件과 充分條件을 各各 구하고, 또 이들의 完全條件은 될 수 없음을 各各 反例를 들어서 밝혔다.

Let R be a ring with unity 1≠0, and M a unital right R-module. M is said to be quasi-projective iff any homorphism f:M->M/N can be lifted to an endomorphism f:M->M such that f=??, where N is a submodule of M and ν is the canonical mapping of M onto the quotient module M/N.

The objective of this note is to find two useful conditions on a ring, namely, a necessary condition and a sufficient condition that every homomorphic image of R be quasi-projective. Also we show that each of the two conditions is not an exact condition for the result with a counter-example respectively.

Let R be a ring with unity 1≠0, and M a unital right R-module. M is said to be quasi-projective iff any homorphism f:M->M/N can be lifted to an endomorphism f:M->M such that f=??, where N is a submodule of M and ν is the canonical mapping of M onto the quotient module M/N.

The objective of this note is to find two useful conditions on a ring, namely, a necessary condition and a sufficient condition that every homomorphic image of R be quasi-projective. Also we show that each of the two conditions is not an exact condition for the result with a counter-example respectively.