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Article
The Bass-Quillen problem on a class of local rings with weak global dimension two
Let (R, m) be a local GCD domain. R is called a U 2 ring if there is an element u ∈ m − m2 such that R/(u) is a valuation domain and R u ...
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Book
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Chapter
Erratum to: Foundations of Commutative Rings and Their Modules
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Chapter
Relative Homological Algebra
Relative homological algebra was conceived by Auslander and Bridger [14, 15] and was formed by Enochs, Jenda, and Torrecillas (for example, [54, 55, 57]).
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Chapter
Basic Theory of Rings and Modules
All the rings in this book are commutative. Sometimes we may also assume that an algebra may be non-commutative.
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Chapter
Homological Methods
To clarify the structure of a ring, homological methods are vry important and effective. Homological methods deal with questions that appear naturally in the category of modules, so category theory is a good s...
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Chapter
Extensions of Rings
This chapter describes some theories of ring extensions, especially of integral extension. The concept of integral dependence, originally proposed by Noether, is a very useful tool for multiplicative ideal the...
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Chapter
Multiplicative Ideal Theory over Integral Domains
Classical ideal theory is based on the work of Krull, Noether, Prüfer and other researchers – these are collated in [68]).
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Chapter
Coherent Rings with Finite Weak Global Dimension
One of the famous results characterized by homological tricks is that a regular local ring is a UFD.
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Chapter
The Grothendieck Group of a Ring
Algebraic K-theory extends many of the methodologies of linear algebra over a number field to over a ring R.
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Chapter
The Category of Modules
Statements and research on module theory and homology theory closely related to the theory and methods of categories, and so the introduction of the concept of some categories is beneficial.
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Chapter
Basic Theory of Noetherian Rings
The class of Noetherian rings has an extremely important significance to geometric applications.
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Chapter
w-Modules over Commutative Rings
In 1977, Glaz and Vasconcelos [73] introduced the concept of semidivisorial modules to study some properties of flat modules.
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Chapter
Structural Theory of Milnor Squares
There are a lot of papers that deal with pullbacks. Milnor squares are actually very important pullbacks in applications. Milnor squares’ success in producing interesting examples has resulted in a good deal o...
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Article
The Rings in Which All Super Finitely Presented Modules are Gorenstein Projective
Let R be a commutative ring. The Gorenstein super finitely presented dimension of R is defined as $$G\hbox {-}\text{ s.gl. }\dim (R)=\...
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Article
Tensor product of projective-like modules
We show that the class of w-split (resp., w-projective, w-invertible, weak w-projective) modules is closed under the tensor product.
