E . e . enochs put forword the concepts of injective ( projective or flat ) ( pre ) cover and ( pre ) envelope in the early 1980s " , a lot of articles have studied existence and uniqueness of such ( pre ) covers and ( pre ) envelopes , the property of their kernels or cokernels , and character many special rings . moreover , if such kind of ( pre ) covers or ( pre ) envelopes exist , we can construct a complete injective ( projective or flat ) resolvent ( called resolution when exact ) and a partial injective ( projective or flat ) resolvent , and if r is a ring , we can study the relationship of its left global dimension l . d ( r ) ( or its weak dimension w ( r ) ) and the properties of syzygies ( or cosyzygies ) of a resolvent ( or resolution ) , and the relationship of its left global dimension l . d ( r ) ( or its weak dimension ) and the exactness of a resolvent ( or resolution ) 自八十年代初e . e . enochs首次提出并研究?jī)?nèi)射(投射、平坦) (預(yù))蓋及內(nèi)射(投射、平坦) (預(yù))包這些概念以來(lái),大批論文研究此類(lèi)包、蓋的存在性、唯一性問(wèn)題以及它們的核、上核的性質(zhì),并據(jù)此刻畫(huà)了一些常見(jiàn)的特殊環(huán);更進(jìn)一步地,當(dāng)此類(lèi)包、蓋存在時(shí),我們可構(gòu)造相應(yīng)的完全投射(平坦、內(nèi)射)預(yù)解式(當(dāng)正合時(shí)稱(chēng)為完全分解式)以及單邊投射(平坦、內(nèi)射)預(yù)解式,研究了環(huán)的左(右)總體維數(shù)、弱維數(shù)與此類(lèi)分解式的合沖模(或上合沖模)的性質(zhì)、復(fù)形正合性之間的關(guān)系。
At first a lot of new characterizations of gorenstein injective modules are given , then the author claim that a ring r is qf if and only if every left ( or right ) r - modules are gorenstein injective , and then show that if r is two - side noetherian , r is n - gorenstein if and only if every n - th cosyzygy of an injective resolution of a left ( and right ) r - module is gorenstein injective if and only if every n - th syzygy of an injective resolvent of a left ( and right ) right module is gorenstein injective . finally , we prove that for an n - gorenstein ring r with n > 0 , every module can be embedded in a gorenstein injective module and the injective dimension of its cokernel is at most n - 1 首先給出了gorenstein內(nèi)射模的許多新的刻畫(huà),推出了環(huán)r是qf環(huán)當(dāng)且僅當(dāng)每個(gè)左(右)的r -模的單邊內(nèi)射分解式的第n個(gè)上合沖是gorenstein內(nèi)射模,接著推出了左、右noether環(huán)只是n - gorenstein環(huán)當(dāng)且僅當(dāng)每個(gè)左(右)模的單邊內(nèi)射分解式的第n個(gè)上合沖是gorenstein內(nèi)射模當(dāng)且僅當(dāng)每個(gè)左(右)模的單邊內(nèi)射預(yù)解式的第n合沖是gorenstein內(nèi)射模,最后推出了n - gorenstein環(huán)中每個(gè)模都可嵌入到一個(gè)gorenstein內(nèi)射模之中,且其上核的內(nèi)射維數(shù)不大于n - 1 。
In the second section , the author studies copure injective modules , which are the kernels of injective precovers . at first the author gives some characterizations of copure injective modules , show many characterizations of reduced copure injective modules , and then study when injective precover is exact . moreover , the author claims that if l . pid ( r ) of a ring is finite , some copure injective modules can be obtained by a resolvent , finally analyze the relationship between syzygies of a resolvent and cosyzygies of a resolution on n - gorenstein rings 第二部分著重研究了上純內(nèi)射模,即內(nèi)射預(yù)蓋的核,首先給出了上純內(nèi)射模的一些等價(jià)刻畫(huà),然后給出了約化的上純內(nèi)射模的等價(jià)刻畫(huà),接著研究了內(nèi)射預(yù)蓋在什么條件下正合,再接著研究了當(dāng)環(huán)的l . pid ( r )有限時(shí)由模的內(nèi)射預(yù)(分)解式可得到一些上純內(nèi)射模,最后討論了n - gorenstein環(huán)中單邊內(nèi)射預(yù)解式的合沖模與單邊內(nèi)射分解式的上合沖模之間的聯(lián)系。
英文解釋
the straight line configuration of 3 celestial bodies (as the sun and earth and moon) in a gravitational system