论文标题
具有双线性形式的代数，并且具有同型内态性
Algebras with a bilinear form, and Idempotent endomorphisms
论文作者
论文摘要
所有$ k $ - 代数都带有双线性形式，其对象都是对$（r，b）$，其中$ r $是$ k $ -algebra和$ b \ b \ b \ colon r \ times r \ times r \ times r \ t t k $是双线性映射，相当于Unital $ k $ -algeberas $ $ $ a $ $ a $ a $ a $ a $ - $（k，1）\ to（a，1_a）unital $ k $ -algebras是$ k $ modules类别的分裂单态。称这种分裂的单态性为代数的左侧对左逆。 $ k $ - 代数的类别具有较弱的增强性的类别和类别的类别，$ k $ -k $ -k $ -sgerbras $（a，b_a）$，b_a $ b_a $兼容与$ a $的乘以$ a $，即$ a $ a $ a $ b_a $ b_a $ a $ $ $ xy = zw $。
The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the canonical homomorphism $(k,1)\to(A,1_A)$ of unital $k$-algebras is a splitting monomorphism in the category of $k$-modules. Call the left inverses of this splitting monomorphism "weak augmentations" of the algebra. There is a category isomorphism between the category of $k$-algebras with a weak augmentation and the category of unital $k$-algebras $(A,b_A)$ with a bilinear form $b_A$ compatible with the multiplication of $A$, i.e., such that $b_A(x,y)=b_A(z,w)$ for all $x,y,z,w\in A$ for which $xy=zw$.
