Journal article
Idealization of a module
Journal of commutative algebra, Vol.1(1), pp.3-56
2009
DOI: 10.1216/JCA-2009-1-1-3
Abstract
Let 𝑅 be a commutative ring and 𝑀 an 𝑅-module. Nagata introduced the idealization 𝑅(+)𝑀 of 𝑀. Here 𝑅(+)𝑀 = 𝑅 ⊕ 𝑀 (direct sum) is a commutative ring with product (𝑟₁, 𝑚₁)(𝑟₂, 𝑚₂) = (𝑟₁𝑟₂, 𝑟₁𝑚₂ + 𝑟₂𝑚₁). The name comes from the fact that if 𝑁 is a submodule of 𝑀, then 0 ⊕ 𝑁 is an ideal of 𝑅(+)𝑀. The idealization can be used to extend results about ideals to modules and to provide interesting examples of commutative rings with zero divisors. We survey known results concerning 𝑅(+)𝑀 and give some new ones too. The theme throughout is how properties of 𝑅(+)𝑀 are related to those of 𝑅 and 𝑀.
Details
- Title: Subtitle
- Idealization of a module
- Creators
- D. D AndersonMichael Winders
- Resource Type
- Journal article
- Publication Details
- Journal of commutative algebra, Vol.1(1), pp.3-56
- DOI
- 10.1216/JCA-2009-1-1-3
- ISSN
- 1939-0807
- eISSN
- 1939-2346
- Language
- English
- Date published
- 2009
- Academic Unit
- Mathematics
- Record Identifier
- 9983985840102771
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