• 호남수학학술지
• /
• 제36권1호
• /
• pp.1-9
• /
• 2014

In this paper, we investigate some properties in commutative diagrams with fiber products of three modules.

• 호남수학학술지
• /
• 제45권3호
• /
• pp.410-417
• /
• 2023

Suppose that there are 8 abelian varieties defined over a number field K which satisfy a commutative diagram. We show that if we know that three out of four short exact sequences satisfy the rate formula of Tate-Shafarevich groups, then the unknown short exact sequence satisfies the rate formula of Tate-Shafarevich groups, too.

• 대한수학회지
• /
• 제60권6호
• /
• pp.1215-1231
• /
• 2023

In this paper, we investigate the nonnil-exact sequences and nonnil-commutative diagrams and show that they behave in a way similar to the classical ones in Abelian categories.

• 대한수학회논문집
• /
• 제11권2호
• /
• pp.285-294
• /
• 1996

In this paper, we show relations between injective covers and direct sums, some commutative properties, and composition properties in the injective covers.

• Korean Journal of Mathematics
• /
• 제14권2호
• /
• pp.217-226
• /
• 2006

In this paper, we investigate the mutual relation among short exact sequences of amalgamated free products which involve augmentation ideals and relation modules. In particular, we find out commutative diagrams having a steady structure in the sense that all of their three columns and rows are short exact sequences.

• Kyungpook Mathematical Journal
• /
• 제64권1호
• /
• pp.1-14
• /
• 2024

Recently, Zhao, Wang, and Pu introduced and studied new concepts of nonnil-commutative diagrams and nonnil-projective modules. They proved that an R-module that is nonnil-isomorphic to a projective module is nonnil-projective, and they proposed the following problem: Is every nonnil-projective module nonnil-isomorphic to some projective module? In this paper, we delve into some new properties of nonnil-commutative diagrams and answer this problem in the affirmative.

• 한국수학사학회지
• /
• 제14권1호
• /
• pp.115-123
• /
• 2001

In this paper we consider the topological properties of $\textit{\Theta}_\textit{\kappa}$and show that the induced map $\tilde{\textit{\Theta}_\textit{\kappa}}$ is well defined and renders the diagram commutative.

• 대한수학회보
• /
• 제33권2호
• /
• pp.303-310
• /
• 1996

The Gottlieb group of a compact connected ANR X, G(X), consists of all $\alpha \in \prod_{1}(X)$ such that there is an associated map $A : S^1 \times X \to X$ and a homotopy commutative diagram $$ S^1 \times X \longrightarrow^A X $$ $$incl \uparrow \nearrow \alpha \vee id $$ $$ S^1 \vee X $$.

• 한국수학교육학회지시리즈A:수학교육
• /
• 제14권1호
• /
• pp.22-26
• /
• 1975

In this paper, we define a differential module and study its properties. In section 2, as for propositions, Ive research some properties, directsum, isomorphism of factorization, exact sequence of derived modules. And then as for theorem, I try to present the following statement, if the sequence of homomorphisms of differential modules is exact. Then the sequence of homomorphisms of Z(X) is exact, also the sequence of homomorphisms of Z(X) is exact. According to the theorem, as for Lemma, we consider commutative diagram between exact sequence of Z(X) and exact sequence of Z'(X) . As an immediate consequence of this theorem, we obtain the following result. If M is an arbitrary module and the sequence of homomorphisms of the modules Z(X) is exact, then the sequence of their tensor products with the trivial endomorphism is semi-exact.