• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.177-185
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• 2011

A right ideal I of a ring R is small in case for every proper right ideal K of R, K + I ${\neq}$ = R. A right R-module M is called PS-injective if every R-homomorphism f : aR ${\rightarrow}$ M for every principally small right ideal aR can be extended to R ${\rightarrow}$ M. A ring R is called right PS-injective if R is PS-injective as a right R-module. We develop, in this article, PS-injectivity as a generalization of P-injectivity and small injectivity. Many characterizations of right PS-injective rings are studied. In light of these facts, we get several new properties of a right GPF ring and a semiprimitive ring in terms of right PS-injectivity. Related examples are given as well.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.251-258
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• 2002

Let R be a commutative Noetherian ring and M an R-module. In this paper we will consider the asymptotic behaviour of ideals relative to an R-module E which is M-injective.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.223-234
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• 2003

This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right YJ-injective, right and left V-ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that $_{A}$A/J is p-injective, then the ring A/J is strongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p.p.ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left YJ-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular : (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, $_{A}$M/Z(M) is projective. ((5) remains valid if 《continuous》 is replaced by 《self-injective》 and 《cyclic》 is replaced by 《finitely generated》. Finally, we have the following two equivalent properties for A to be von Neumann regula. : (a) A is left non-singular such that every finitely generated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change 《left non-singular》 into 《right non-singular》in (a).(a).

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.299-307
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• 2006

The following results ale extended from P-injective rings to AP-injective rings: (1) R is left self-injective regular if and only if R is a right (resp. left) AP-injective ring such that for every finitely generated left R-module M, $_R(M/Z(M))$ is projective, where Z(M) is the left singular submodule of $_{R}M$; (2) if R is a left nonsingular left AP-injective ring such that every maximal left ideal of R is either injective or a two-sided ideal of R, then R is either left self-injective regular or strongly regular. In addition, we answer a question of Roger Yue Chi Ming [13] in the positive. Let R be a ring whose every simple singular left R-module is Y J-injective. If R is a right MI-ring whose every essential right ideal is an essential left ideal, then R is a left and right self-injective regular, left and right V-ring of bounded index.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.395-409
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• 2012

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let ${\pi}$ be a group and let M ${\rightarrow}$ N be a homomorphism between projective $\mathbb{Z}[{\pi}]$-modules such that $\mathbb{Z}_p\;{\otimes}_{\mathbb{Z}[{\pi}]}M{\rightarrow}\mathbb{Z}_p{\otimes}_{\mathbb{Z}[{\pi}]}\;N$ is injective; for which other right $\mathbb{Z}[{\pi}]$-modules V is the induced map $V{\otimes}_{\mathbb{Z}[{\pi}]}\;M{\rightarrow}\;V{\otimes}_{\mathbb{Z}[{\pi}]}\;N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.

• Kyungpook Mathematical Journal
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• v.62 no.4
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• pp.641-655
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• 2022

A right R-module N is called pseudo semisimple-M-injective if for any monomorphism from every semisimple submodule of M to N, can be extended to a homomorphism from M to N. In this paper, we study some properties of pseudo semisimple-injective modules. Moreover, some results of pseudo semisimple-injective modules over formal triangular matrix rings are obtained.

• Korean Journal of Mathematics
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• v.4 no.1
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• pp.45-49
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• 1996

Let $F_0$ be the maximal real subfield of $\mathbb{Q}({\zeta}_q+{\zeta}_q^{-1})$ and $F_{\infty}={\cup}_{n{\geq}0}F_n$ be its basic $\mathbb{Z}_p$-extension. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $F_n$. The aim of this paper is to examine the injectivity of the natural $mapA_n{\rightarrow}A_m$ induced by the inclusion $F_n{\rightarrow}F_m$ when $m>n{\geq}0$. By using cyclotomic units of $F_n$ and by applying cohomology theory, one gets the following result: If $p$ does not divide the order of $A_1$, then $A_n{\rightarrow}A_m$ is injective for all $m>n{\geq}0$.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.611-622
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• 2010

A ring R is called left max-coherent provided that every maximal left ideal is finitely presented. $\mathfrak{M}\mathfrak{I}$ (resp. $\mathfrak{M}\mathfrak{F}$) denotes the class of all max-injective left R-modules (resp. all max-flat right R-modules). We prove, in this article, that over a left max-coherent ring every right R-module has an $\mathfrak{M}\mathfrak{F}$-preenvelope, and every left R-module has an $\mathfrak{M}\mathfrak{I}$-cover. Furthermore, it is shown that a ring R is left max-injective if and only if any left R-module has an epic $\mathfrak{M}\mathfrak{I}$-cover if and only if any right R-module has a monic $\mathfrak{M}\mathfrak{F}$-preenvelope. We also give several equivalent characterizations of MI-injectivity and MI-flatness. Finally, $\mathfrak{M}\mathfrak{I}$-dimensions of modules and rings are studied in terms of max-injective modules with the left derived functors of Hom.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.439-450
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• 2019

Let R be a ring with unity. Given modules $M_R$ and $_RN$, $M_R$ is said to be absolutely $_RN$-pure if $M{\otimes}N{\rightarrow}L{\otimes}N$ is a monomorphism for every extension $L_R$ of $M_R$. For a module $M_R$, the subpurity domain of $M_R$ is defined to be the collection of all modules $_RN$ such that $M_R$ is absolutely $_RN$-pure. Clearly $M_R$ is absolutely $_RF$-pure for every flat module $_RF$, and that $M_R$ is FP-injective if the subpurity domain of M is the entire class of left modules. As an opposite of FP-injective modules, $M_R$ is said to be a test for flatness by subpurity (or t.f.b.s. for short) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. Every ring has a right t.f.b.s. module. $R_R$ is t.f.b.s. and every finitely generated right ideal is finitely presented if and only if R is right semihereditary. A domain R is $Pr{\ddot{u}}fer$ if and only if R is t.f.b.s. The rings whose simple right modules are t.f.b.s. or injective are completely characterized. Some necessary conditions for the rings whose right modules are t.f.b.s. or injective are obtained.

• Tunnel and Underground Space
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• v.25 no.4
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• pp.320-331
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• 2015

This paper presents a case study of an enhanced geothermal system(EGS) demonstration project conducted in $Gro{\ss}$ $Sch{\ddot{o}}nebeck$, Northerm Germany, focusing on hydraulic stimulation. The project was conducted with doublet system in sandstone and volcanic formations at 4 - 4.4 km depth. Under normal faulting to strike-slip faulting stress regime, hydraulic stimulations were conducted at injection and production wells by massive waterfrac and gel-proppant fracturing. Injectivity index increased from $0.97m^3/(hr^*MPa)$ to $7.5m^3/(hr^*MPa)$ and productivity index increased from $2.4m^3/(hr^*MPa)$ to $10.1m^3/(hr^*MPa)$ by a series of hydraulic stimulations at both wells. After circulation tests through injection and production wells, however, productivity index decreased from $8.9m^3/(hr^*MPa)$ to $0.6m^3/(hr^*MPa)$ in two years. Slip tendency analysis for the stimulation in volcanic layer estimated the required pressure for shear slip and its preferred orientations and it showed reasonable match with actual stimulation results. Through the microseismicity observation for the stimulation of volcanic formation, only 80 seismic events with its moment magnitudes in -1.8<$M_W$<-1.0 were observed, which are unexpectedly low for EGS hydraulic stimulation.