What is the meaning of this notation $\mathbb C_2$ in Jacobson's Basic Algebra I?

Question

I was looking at Chapter 2 (Rings) in Basic Algebra I by Jacobson (1974). (Section 2.1 is Definition and Elementary Properties, 2.2 is Types of Rings, 2.3 is Matrix Rings (the ring $M_n(R)$ of $n\times n$ matrices over the ring $R$), and 2.4 is Quaternions.)

On page 96, Jacobson defines quaternions:

We consider the subset $\mathbb H$ of the ring $M_2(\mathbb C)$ of... matrices that have the form... $\pmatrix{a&b\\-\overline b&\overline a}$. We claim that $\mathbb H$ is a subring of $\color{blue}{\mathbb C_2}$...$\mathbb H$ is a subgroup of the additive group of $M_2(\mathbb C)$. We obtain the unit matrix by taking $a=1, b=0$.... $\mathbb H$ is closed under multiplication and so $\mathbb H$ is a subring of $M_2(\mathbb C)$. ... Every non-zero element of $\mathbb H$ has an inverse in $\color{blue}{\mathbb C_2}$... and... it is contained in $\mathbb H$. Hence $\mathbb H$ is a division ring.

My question is what is $\color{blue}{\mathbb C_2}$? Is it a typographical error for $M_2(\mathbb C)$, or a synonym for $M_2(\mathbb C)$? I don't see it defined in the text, but I see it twice on the page. (I figure a two-dimensional vector space over $\mathbb C$ would be written $\mathbb C^2$ (superscript $2$).)

Answer 1

I expect that this is a typo. On page 98 of the second edition of the text (published 1985), it has "$M_2(\mathbb{C})$" where you first have "$\mathbb{C}_2$":

and again on page 99, where you have $\mathbb{C}_2$ again, it has "$M_2(\mathbb{C})$" again:

Answer 2

I don't have the book you mention, so I'm only answering based on the provided context.

Going by the definition (e.g., see this section), it looks like $\mathbb{C}_2$ is indeed a typo or an ad-hoc abbreviation for $M_2(\mathbb{C})$; if the book has a note on notational conventions at the beginning, that should tell you which one it is.

$M_n({R})$ is the ring of $n \times n$ matrices over the ring $R$. That makes $M_2(\mathbb{C})$ the ring of all $2 \times 2$ matrices over the ring of complex numbers.