A question about a ring in the form of a direct product.

Question

Atiyah-Macdonald's "Introduction to Commutative Algebra" (Pg. 20) says the following:

Suppose that the ring $A$ is a direct product $\Pi_{i=1}^n A_i$. Then the set of all elements of $A$ of the form $$(0,0,\dots,0,a_i,0,\dots0,0)$$ with $a\in A_i$ is an ideal $a_i$ of $A$. The ring $A$ is the direct sum of the ideals $a_1,a_2,\dots,a_n$

1. Are $A_1,A_2,\dots,A_n$ all rings?

2. The ideal $a_1$ of $A$ is of the form $(a_1,0,0,\dots,0)$. Similarly, $a_2$ is of the form $(0,a_2,0,0,\dots,0)$. What is $a_1\oplus a_2$? Shouldn't it be of the form $(a_1,0,\dots,0,0,a_2,\dots,0)$, which is a $2n$-tuple?

3. How is the product of two $n$-tuples defined? Wouldn't it need to be defined for $A$ to be a ring? Is it component-wise?

Thank you.

Answer 1

1) Yes, all the $A_i$ are rings.

2) Here the term "direct sum" means internal direct sum, which basically just means "sum", but with the additional property that the ideals intersect trivially.

3) Yes, the product is component-wise (just like the sum).

Answer 2

1. Yes

2. and 3.: Addition and multiplication is defined component-wise. So $a_1 \oplus a_2 = (a_1, a_2, 0, ..., 0)$.