Which of the following rings are integral domains?
(a) $\mathbb{R}[x]$, the ring of all polynomials in one variable with real coefficients.
(b) $M_n(\mathbb{R}) $.
(c) the ring of complex analytic functions defined on the unit disc of the complex plane (with pointwise addition and multiplication as the ring operations).
Only (a) & (c) are correct. Am I right?
So for (a) is is true that whenever $D$ is an integral domain, $D[x]$ is also an integral domain. To see this, when you multiply two polynomials, look at the leading term of the product and observe that it has to be nonzero if you started with two nonzero polynomials.
On the other hand, (b) only holds for $n>1$, because $M_1(\mathbb R)$ is isomorphic to $\mathbb R$. For $n>1$, you can just do something silly like multiply a matrix with a $1$ in the upper left hand corner by a matrix with a $1$ in the bottom right hand corner and $0$'s elsewhere. Notice this breaks down for $n=1$.
(c) Follows because nonzero complex analytic functions have isolated zeros. Thus if you take two nonzero complex analytic functions on the unit disk, their zero sets are both finite in the disk, so there's a point in the disk where they both don't vanish, and their product won't vanish there either.
edit: For (c), to avoid the annoyance of zeroes accumulating at the boundary, we can just say that away from the boundary, for example, in the circle of radius $\frac{1}{2}$, this function definitely only has finitely many zeros, and then we can conclude.
