Determine the cardinality of set of subrings of $\mathbb{Q}$. (Hint: For a set $P$ of positive prime numbers, consider the smallest subring of $\mathbb{Q}$ that contains $\left\{\left.\frac{1}{p}\right\rvert\, p \in P\right\}$.)

Let $R$ be a commutative ring. The characteristic of $R$ is the smallest positive integer $n$ such that $a + a + \cdots + a$ ($n$ times) is zero for every $a \in R$, if such an integer exists, and zero, otherwise. Choose the correct statement(s) from below:
(A) For every $n \in \mathbb{N}$, there exists a commutative ring whose characteristic is $n$;
(B) There exists an integral domain with characteristic 57;
(C) The characteristic of a field is either 0 or a prime number;
(D) For every prime number $p$, every commutative ring of characteristic $p$ contains $\mathbb{F}_p$ as a subring.

Let $F$ be a field and $R$ a subring of $F$ that is not a field. Let $x$ be a variable. Let $S = \left\{ a _ { 0 } + a _ { 1 } x + \cdots + a _ { n } x ^ { n } \mid n \geq 0 \text{ and } a _ { 0 } \in R , a _ { 1 } , \cdots a _ { n } \in F \right\}$.
(A) (2 marks) Show that, with the natural operations of addition and multiplication of polynomials, $S$ is an integral domain.
(B) (4 marks) Let $I = \{ f ( x ) \in S \mid f ( 0 ) = 0 \}$. Determine whether $I$ is a prime ideal.
(C) (4 marks) Determine whether $S$ is a PID.

Which of the following statement(s) are true?
(A) Every prime ideal of a finite commutative ring with unity is maximal.
(B) A commutative ring with unity whose set of all ideals is countably infinite is necessarily a countable ring.
(C) Let $R$ be a unique factorisation domain and $K$ be its field of fractions. There exists an irreducible element $\alpha \in R$ and an element $\beta \in K$ such that $\beta ^ { 2 } = \alpha$.
(D) Every subring $R$ (with unity) of $\mathbb { Q }$ with $\mathbb { Z } \varsubsetneqq R \varsubsetneqq \mathbb { Q }$ has infinitely many prime ideals.

Show that $\mathscr { D } _ { \rho } ( \mathbb { R } )$ is an integral domain.

An operation $\Theta$ is defined on the set of integers for every integers a and b as
$$a \ominus b = a - b + 1$$
Regarding the $\Theta$ operation, I. The identity element is 1. II. It has the commutative property. III. It has the associative property. Which of these statements are true?
A) Only I
B) I and II
C) I and III
D) II and III
E) I, II and III