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- 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 , defined by taking a (finite) sum to (Here and .) In less formal terms, for any element , for any element , and is then extended to a homomorphism of R-modules in the obvious way. The augmentation ideal A is the kernel of and is therefore a two-sided ideal in R[G]. A is generated by the differences of group elements. Equivalently, it is also generated by , which is a basis 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. The augmentation ideal plays a basic role in group cohomology, amongst other applications. (en)
- 代數中，增廣理想是可以在任何群環中定義的一種理想。若G是群，R是交換環，則有一個自群環R[G]至R的環同態，稱為增廣映射，將R[G]的元素 映射至 其中ri是R的元素，gi是G的元素。按照群環的定義，以上的和是有限和。較籠統而言，對G任何元素g，定義 為1R，再將以最顯然的方法延伸成R-模的同態。增廣理想是的核，因此是R[G]的雙邊理想，由群元素的差 生成。 此外，增廣理想是自由R-模，可用 為其基底而生成。 對上述的R和G，群環R[G]是增廣R-代數的一例。這樣的代數都帶有一個映至R上的環同態。這個環同態的核是這個代數的增廣理想。 增廣理想的另一類例子是任何霍普夫代數的的核。 增廣理想是群上同調等應用中的基本工具。 (zh)
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- 代數中，增廣理想是可以在任何群環中定義的一種理想。若G是群，R是交換環，則有一個自群環R[G]至R的環同態，稱為增廣映射，將R[G]的元素 映射至 其中ri是R的元素，gi是G的元素。按照群環的定義，以上的和是有限和。較籠統而言，對G任何元素g，定義 為1R，再將以最顯然的方法延伸成R-模的同態。增廣理想是的核，因此是R[G]的雙邊理想，由群元素的差 生成。 此外，增廣理想是自由R-模，可用 為其基底而生成。 對上述的R和G，群環R[G]是增廣R-代數的一例。這樣的代數都帶有一個映至R上的環同態。這個環同態的核是這個代數的增廣理想。 增廣理想的另一類例子是任何霍普夫代數的的核。 增廣理想是群上同調等應用中的基本工具。 (zh)
- 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 , defined by taking a (finite) sum to (Here and .) In less formal terms, for any element , for any element , and is then extended to a homomorphism of R-modules in the obvious way. The augmentation ideal A is the kernel of and is therefore a two-sided ideal in R[G]. The augmentation ideal plays a basic role in group cohomology, amongst other applications. (en)
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- Augmentation ideal (en)
- 增廣理想 (zh)
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