• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1221-1233
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• 2018

Let R be a commutative ring with non-zero identity and let NN(R) = {I | I is a nonnil ideal of R}. Let M be an R-module and let ${\phi}-tor(M)=\{x{\in}M{\mid}Ix=0\text{ for some }I{\in}NN(R)\}$. If ${\phi}or(M)=M$, then M is called a ${\phi}$-torsion module. An R-module M is said to be ${\phi}$-flat, if $0{\rightarrow}{A{\otimes}_R}\;{M{\rightarrow}B{\otimes}_R}\;{M{\rightarrow}C{\otimes}_R}\;M{\rightarrow}0$ is an exact R-sequence, for any exact sequence of R-modules $0{\rightarrow}A{\rightarrow}B{\rightarrow}C{\rightarrow}0$, where C is ${\phi}$-torsion. In this paper, the concepts of NRD-submodules and NP-submodules are introduced, and the ${\phi}$-flat modules over a ${\phi}-Pr{\ddot{u}}fer$ ring are investigated.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1513-1528
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• 2021

Let R be a commutative ring with prime nilradical Nil(R) and M an R-module. Define the map 𝜙 : R → R_Nil(R) by ${\phi}(r)=\frac{r}{1}$ for r ∈ R and 𝜓 : M → M_Nil(R) by ${\psi}(x)=\frac{x}{1}$ for x ∈ M. Then 𝜓(M) is a 𝜙(R)-module. An R-module P is said to be 𝜙-projective if 𝜓(P) is projective as a 𝜙(R)-module. In this paper, 𝜙-exact sequences and 𝜙-projective R-modules are introduced and the rings over which all R-modules are 𝜙-projective are investigated.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2043-2051
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• 2017

Consider a sequence of compactly supported Hamiltonian diffeomorphisms ${\phi}_k$ of an exact symplectic manifold, all of which are "graphical" in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the ${\phi}_k$ $C^0$-converge to the identity, then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the ${\phi}_k$. We discuss connections to the open problem of whether the Calabi homomorphism extends to the Hamiltonian homeomorphism group. The proof is based on a relationship between the Calabi invariant of a $C^0$-small Hamiltonian diffeomorphism and the generalized phase function of its graph.