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- Augmentation_ideal abstract "In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If G is a group and R a commutative ring, there is a ring homomorphism , called the augmentation map, from the group ring to R, defined by taking a sum to Here ri is an element of R and gi an element of G. The sums are finite, by definition of the group ring. In less formal terms, is defined as 1R whatever the element g in G, and is then extended to a homomorphism of R-modules in the obvious way. The augmentation ideal is the kernel of , and is therefore a two-sided ideal in R[G]. It is generated by the differences of group elements.Furthermore it is also generated by which is a basis for the augmentation ideal as a free R module.For R and G as above, the group ring R[G] is an example of an augmented R-algebra. Such an algebra comes equipped with a ring homomorphism to R. The kernel of this homomorphism is the augmentation ideal of the algebra.Another class of examples of augmentation ideal can be the kernel of the counit of any Hopf algebra.The augmentation ideal plays a basic role in group cohomology, amongst other applications.".
- Augmentation_ideal wikiPageID "2224774".
- Augmentation_ideal wikiPageRevisionID "559286481".
- Augmentation_ideal hasPhotoCollection Augmentation_ideal.
- Augmentation_ideal subject Category:Hopf_algebras.
- Augmentation_ideal subject Category:Ideals.
- Augmentation_ideal type Abstraction100002137.
- Augmentation_ideal type Algebra106012726.
- Augmentation_ideal type Cognition100023271.
- Augmentation_ideal type Content105809192.
- Augmentation_ideal type Discipline105996646.
- Augmentation_ideal type HopfAlgebras.
- Augmentation_ideal type Idea105833840.
- Augmentation_ideal type Ideal105923696.
- Augmentation_ideal type Ideals.
- Augmentation_ideal type KnowledgeDomain105999266.
- Augmentation_ideal type Mathematics106000644.
- Augmentation_ideal type PsychologicalFeature100023100.
- Augmentation_ideal type PureMathematics106003682.
- Augmentation_ideal type Science105999797.
- Augmentation_ideal comment "In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If G is a group and R a commutative ring, there is a ring homomorphism , called the augmentation map, from the group ring to R, defined by taking a sum to Here ri is an element of R and gi an element of G. The sums are finite, by definition of the group ring. In less formal terms, is defined as 1R whatever the element g in G, and is then extended to a homomorphism of R-modules in the obvious way.".
- Augmentation_ideal label "Augmentation ideal".
- Augmentation_ideal label "增廣理想".
- Augmentation_ideal sameAs m.06x9h6.
- Augmentation_ideal sameAs Q4820423.
- Augmentation_ideal sameAs Q4820423.
- Augmentation_ideal sameAs Augmentation_ideal.
- Augmentation_ideal wasDerivedFrom Augmentation_ideal?oldid=559286481.
- Augmentation_ideal isPrimaryTopicOf Augmentation_ideal.