Let $M$ be a matrix of $M _ { n } ( A )$. The purpose of this question is to generalize to an arbitrary commutative ring $A$ the two formulas recalled in the introduction when $A$ is a field. a) Show that if the ring $A$ is integral, then $M \widetilde { M } = \widetilde { M } M = ( \operatorname { det } M ) I _ { n }$. b) We no longer assume $A$ is integral. Show that the result of a) still holds if there exists a surjective ring morphism $B \rightarrow A$ with $B$ integral. c) Deduce that the result of a) still holds for every commutative ring $A$. d) Prove that if $M$ and $N$ are in $M _ { n } ( A )$, then we have $$\operatorname { det } ( M N ) = \operatorname { det } M \times \operatorname { det } N .$$

A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a, b \in W$, the numbers $a - b$ and $ab$ are also in $W$. Let $S = \left\{ \left. \frac{m}{2^n} \right\rvert\, m, n \text{ integers} \right\}$ and $T = \left\{ \left. \frac{p}{q} \right\rvert\, p, q \text{ integers}, q \text{ odd} \right\}$. Then
(A) neither $S$ nor $T$ is a ring
(B) $S$ is a ring, $T$ is not a ring
(C) $T$ is a ring, $S$ is not a ring
(D) both $S$ and $T$ are rings

A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then:
(a) neither $S$ nor $T$ is a ring
(b) $S$ is a ring, $T$ is not a ring.
(c) $T$ is a ring, $S$ is not a ring.
(d) both $S$ and $T$ are rings.

A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then:
(a) neither $S$ nor $T$ is a ring
(b) $S$ is a ring, $T$ is not a ring.
(c) $T$ is a ring, $S$ is not a ring.
(d) both $S$ and $T$ are rings.

A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a, b \in W$, the numbers $a - b$ and $ab$ are also in $W$. Let $S = \left\{ \left. \frac{m}{2^n} \right\rvert\, m, n \text{ integers} \right\}$ and $T = \left\{ \left. \frac{p}{q} \right\rvert\, p, q \text{ integers}, q \text{ odd} \right\}$. Then
(A) neither $S$ nor $T$ is a ring
(B) $S$ is a ring, $T$ is not a ring
(C) $T$ is a ring, $S$ is not a ring
(D) both $S$ and $T$ are rings

A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then
(A) neither $S$ nor $T$ is a ring
(B) $S$ is a ring, $T$ is not a ring
(C) $T$ is a ring, $S$ is not a ring
(D) both $S$ and $T$ are rings