Algebraic Torus: Etymology

Question

I was wondering if anyone might be able to tell me why algebraic tori are called specifically algebraic tori? I find it difficult to see exactly how "an algebraic group that can be described as a direct product of finitely many multiplicative groups" is in any sense similar to the more standard notion of a doughnut.

Any and all help is appreciated.

Answer 1

Well, seeing this question recently received an upvote, I feel that perhaps I should label this as answered. Specifically, back when they made the comment in question, I found leibnewtz' statement on the matter to satisfy me entirely. Thus, in bigger letters:

The algebraic $n$-torus $(\mathbb{C}^{*})^n$ is homotopy-equivalent to the topological $n$-torus $(\mathbb{S}^1)^n$. From here it's easy to generalize to arbitrary rings $R$.

Answer 2

Consider $S^1×S^1.$ This is a torus, and the product of two multiplicative groups (the complex numbers of modulus $1.$)