Generalized 2-Absorbing Ideals On Hochschild Extensions And N-Trivial Extensions

The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory,representation theory,category theory and homological algebra.Ideals play an important role,the property of ideals determines all homomorphism of rings.Therefore,for the ideal of rings,and more importantly,for the ideal of trivial ring extensions,some people have done in-depth research.In this paper,we study generalized 2-absorbing ideals over Hochschild extension and n-trivial extension,which are more extensive than trivial ring extension.In particular,when Hochschild 2-cocycle ? = 0?,Hochschild extension is trivial extension:when n = 1?,1-trivial extension is trivial extension.In this paper,we research on general-ized 2-absorbing ideals over Hochschild extension,including 2-absoring ideals,weakly 2-absorbing ideals,2-absorbing primary ideals,weakly 2-absorbing primary ideals,semi 2-absorbing ideals,n-absorbing ideals and(n,n)-closed ideals,and we get some results when R is special rings,such as local rings,integral domain,Prufer domain,commuta-tive divided rings and reduced commutative rings.In addition,we prove the properties of 2-absorbing ideals,semi 2-absorbing ideals,m-absorbing ideals and(m,s)-closed ideals on n-trivial extension,which extend the property of ideals on trivial extension.The details are as follows:In Chapter 1,we summarize the background of the thesis,and recall the newest development of the literature.Some related definitions and conclusions about general-ized 2-absorptive ideals are introduced.The concept of trivial extensions,some prop-erties of ideals on trivial extensions and the concept of Hochschild extensions are re-viewed.In this paper,we assume R is a commutative rings with nonzero identity.In the preliminary knowledge,the properties of Hochschild 2-cyclica are studied,and we get?(q,r2r3)= ?(1,r2)r3,r1?(1,r3)= ?(r1,1)r3,?(r1r2,1)= r1?(r2,1).On this basis,the hypothesis of this paper ?(a,b)= a?(a,b)is given.For further proof,we first prove the following important lemma:Suppose R is a commutative ring containing non-zero units,I is the proper ideal of R and,is the P-primary ideal of R where(?)I = P is the prime ideal of R.Let M be a R-R-bimodule and N be a submodule of M and IM(?)N,then(1)(?)H?(I,N)= H?((?)I,M);(2)H?(I,N)is a H?((?)I,M)-primary ideal of H?(R,M).The properties of Hochschild extension of rings,such as Jacobson basis,nilpotent ele-ment,ideal height,dimension and zero factor,are also studied.In Chapter 2,we study the properties of generalized 2-absorptive ideals on Hochschild expansion,including 2-absorbing ideals,weak 2-absorbing ideals,2-absorbing primary ideals,weak 2-absorbing primary ideals semi 2-absorbing ideals,n-absorbing ideals and(m,n)-closed ideals,and the properties of related ideals on trivial extensions are general-ized:Assume that R is a commutative ring containing non-zero units,M is a R-R-bimodule,I is the proper ideal of R,for every Hochschild 2-cocycle a,H?(R,M)is the Hochschild extension of R and M,we find that H?(I,M)is a 2-absorbing ideal(semi 2-absorbing ide-al?n-absorbing ideal?(m,n)-closed ideal)of H?(R,M)if and only if I is a 2-absorbing ideal(semi 2-absorbing ideal?n-absorbing ideal?(m,n)-closed ideal)of R.In addition,we also study weakly 2-absorbing ideals,2-absorbing primary ideals and weakly 2-absorbing primary ideals on Hochschild extension and properties of H?(I,N),where N is a submoudle of M,when Hochshild 2-cocycle satisfiesd ?(a,b)= a?(1,b),(?)Va,b? R.we get some results when R is special rings,such as local rings,integral domain,Prufer domain,commutative divided rings and reduced commutative rings.In Chapter 3,we study properties of 2-absorbing ideals,semi 2-absorbing ideals,m-absorbing ideals and(m,s)-closed ideals on n-trivial extension.So the properties of related ideals on trivial extension are further generalized:Assume R:=(Ri)i=1 n is a ring family,M:=(Mi,j)i?<j?n module family,such that for(?)1?i<j?n,Mi,j is a(Ri,Rj)-bimodule,and Ii is a proper ideal of Ri,I:=(Ii)i=1 n,We conclude that Tn(I,M)is a(m,s)-closed ideal(semi 2-absorbing ideal)of Tn(R,M)if and only if Ii is a(m,s)-closed ideal(semi 2-absorbing ideal)of Ri.Assume R:=(Ri)i=1 n is a ring family,M:=(Mi,j)1?i<j?n a module family,such that for(?)1?i<j?n,Mi,j is a(Ri,Rj)-bimodule,and Ii is a proper ideal of Ri,I:=(Ii)i=1 n.We conclude that if Tn(I,M)is a 2-absorbing ideal(m-absorbing ideal)of Tn(R,M),then Ii is a 2-absorbing ideal(m-absorbing ideal)of Ri,but the converse is not true.