# Weakly reflexive algebra vs proper (residually finite-dimensional) algebra

Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine):

An algebra $$A$$ is called proper (or residually finite dimensional) if $$i_A:A \to A^{\circ*}$$ is injective, it is called weakly reflexive if $$i_A$$ is injective, and it is called reflexive if $$i_A$$ is bijective.

This definition is stated as if being proper and weakly reflexive are two different things, but according to their definition these two concepts are equivalent. I'm wondering if there is a typo in the text. Or the two concepts are actually identical?

Here $$A$$ is assumed to be a unital, associative algebra over some field $$k$$. $$A^\circ = \Big\{ f \in A^* : \exists I : \text{Ideal of} \; A \; \text{such that } \; I \subset \ker f \; \text{and}\; \dim_k \frac{A}{I} < \infty \Big\}$$ is a finite-dual coalgebra and $$i_A$$ is a canonical map defined by $$i_A(a)(f) = f(a)$$. The ideal $$I$$ in the above definition is assumed to be two-sided.

I don't want to learn wrong definitions, so I will be grateful if somebody more experienced than me can tell me if the definition is correct.

• Do you assume $A$ commutative? could you mention the book's author? – YCor Oct 15 '19 at 5:27
• No I don't assume $A$ to be commutative. However, I assume $A$ to be unital and associative algebra over some field $k$. – Nik Pronko Oct 15 '19 at 5:29
• Guessing it’s a typo and that the second “injective” was intended to read “surjective.” – Qiaochu Yuan Oct 15 '19 at 5:47
• @QiaochuYuan Thank you. I think I can use your guess as a working hypothesis. – Nik Pronko Oct 15 '19 at 6:02
• The relevant page (which I have also edited into the post): books.google.com/books?id=pBJ6sbPHA0IC&pg=PA44 . – LSpice Oct 15 '19 at 7:33