Let $R$ be a euclidean domain that is not a field. Let $d : R \backslash \{ 0 \} \longrightarrow \mathbb { N }$ be the euclidean size (degree) function. Write $R ^ { \times }$for the invertible elements of $R$. Pick the correct statements from below. (A) $R = R ^ { \times } \cup \{ 0 \}$. (B) There exists $a \in R \backslash \left( R ^ { \times } \cup \{ 0 \} \right)$ such that $d ( a ) = \inf \left\{ d ( r ) \mid r \in R \backslash \left( R ^ { \times } \cup \{ 0 \} \right) \right\}$. (C) With $a$ defined as above, for all $r \in R$, there exists $u \in R ^ { \times } \cup \{ 0 \}$ such that $a$ divides ( $r - u$ ). (D) With $a$ defined as above, the ideal generated by $a$ is a maximal ideal.
Let $R$ be a euclidean domain that is not a field. Let $d : R \backslash \{ 0 \} \longrightarrow \mathbb { N }$ be the euclidean size (degree) function. Write $R ^ { \times }$for the invertible elements of $R$. Pick the correct statements from below.\\
(A) $R = R ^ { \times } \cup \{ 0 \}$.\\
(B) There exists $a \in R \backslash \left( R ^ { \times } \cup \{ 0 \} \right)$ such that $d ( a ) = \inf \left\{ d ( r ) \mid r \in R \backslash \left( R ^ { \times } \cup \{ 0 \} \right) \right\}$.\\
(C) With $a$ defined as above, for all $r \in R$, there exists $u \in R ^ { \times } \cup \{ 0 \}$ such that $a$ divides ( $r - u$ ).\\
(D) With $a$ defined as above, the ideal generated by $a$ is a maximal ideal.