$R=(\{0,1,2,3,4,5\},+_{6}, imes_{6})$ is a :

Question

$R=(\{0,1,2,3,4,5\},+_{6},	imes_{6})$ is a :

Accepted Answer

To determine why Option 1 is the correct answer, we need to analyze the structure $ R = (\{0,1,2,3,4,5\}, +_6, \times_6) $.

### Explanation for Option 1: Ring with Zero Divisors

1. **Ring Structure**: 
   - The set $\{0,1,2,3,4,5\}$ with operations $+_6$ and $\times_6$ is a ring. The operation $+_6$ is addition modulo 6, and $\times_6$ is multiplication modulo 6.
   - A ring is a set equipped with two binary operations satisfying properties such as closure, associativity, distributivity, and the existence of an additive identity (0 in this case).

2. **Zero Divisors**:
   - Zero divisors are elements $a$ and $b$ in a ring such that $a \neq 0$, $b \neq 0$, and $a \times b = 0$.
   - In $R$, consider $2 \times_6 3 = 6 \equiv 0 \pmod{6}$. Here, both 2 and 3 are non-zero elements whose product is zero, indicating the presence of zero divisors.

Thus, $R$ is a ring with zero divisors, confirming Option 1 as correct.

### Explanation for Why Other Options are Incorrect

2. **Field**:
   - A field is a commutative ring with no zero divisors and every non-zero element has a multiplicative inverse.
   - Since $R$ has zero divisors (as shown above), it cannot be a field.

3. **Division Ring**:
   - A division ring is a ring in which division is possible (every non-zero element has a multiplicative inverse), but multiplication may not be commutative.
   - Again, the presence of zero divisors in $R$ disqualifies it from being a division ring, as division by zero divisors is not defined.

4. **Ring without Zero Divisors**:
   - This option suggests that the ring has no zero divisors, which contradicts our earlier finding that $2 \times_6 3 = 0$.
   - Therefore, $R$ cannot be a ring without zero divisors.

In conclusion, the structure $R$ is best described as a ring with zero divisors, making Option 1 the correct choice.