The question asks about the properties of the ring \(R = (Z_2 \times Z_2, +,.)\), where the elements are ordered pairs \((a, b)\) with \(a, b \in Z_2 = \{0, 1\}\). The operations are defined as:

• Addition: \((a, b) + (c, d) = (a + c, b + d)\)
• Multiplication: \((a, b) \cdot (c, d) = (a \cdot c, b \cdot d)\)

All operations \(+\) and \(\cdot\) on the components are performed modulo 2.

Checking for Commutativity of Ring R

A ring \(R\) is commutative if its multiplication is commutative. We need to check if \((a, b) \cdot (c, d) = (c, d) \cdot (a, b)\) for all \((a, b), (c, d) \in R\).

According to the definition:

\((a, b) \cdot (c, d) = (a \cdot c, b \cdot d)\)

\((c, d) \cdot (a, b) = (c \cdot a, d \cdot b)\)

Since multiplication in \(Z_2 = \{0, 1\}\) is commutative (i.e., \(a \cdot c = c \cdot a\) and \(b \cdot d = d \cdot b\) modulo 2), we have \((a \cdot c, b \cdot d) = (c \cdot a, d \cdot b)\).

Thus, \((a, b) \cdot (c, d) = (c, d) \cdot (a, b)\). The multiplication in \(R\) is commutative. Therefore, \(R\) is a commutative ring.

Finding the Unity Element in Ring R

A ring \(R\) has unity if there exists an element \(e \in R\) such that \(e \cdot x = x \cdot e = x\) for all \(x \in R\). This element is the multiplicative identity.

Let the unity element be \((e_1, e_2) \in R\). We need \((a, b) \cdot (e_1, e_2) = (a, b)\) for all \((a, b) \in R\).

Using the multiplication rule: \((a \cdot e_1, b \cdot e_2) = (a, b)\).

This implies \(a \cdot e_1 = a\) (mod 2) and \(b \cdot e_2 = b\) (mod 2) for all \(a, b \in \{0, 1\}\).

In \(Z_2\), the only element \(x\) such that \(y \cdot x = y\) for all \(y \in \{0, 1\}\) is \(x = 1\). So, \(e_1 = 1\) and \(e_2 = 1\).

The element \((1, 1)\) is in \(R\) since \(1 \in Z_2\). Let's verify:

\((a, b) \cdot (1, 1) = (a \cdot 1, b \cdot 1) = (a, b)\)

\((1, 1) \cdot (a, b) = (1 \cdot a, 1 \cdot b) = (a, b)\)

So, \((1, 1)\) is the unity element in \(R\). Therefore, \(R\) is a ring with unity.

Identifying Divisors of Zero in Ring R

A non-zero element \(x\) in a ring \(R\) is a divisor of zero if there exists a non-zero element \(y \in R\) such that \(x \cdot y = 0\). Here, \(0\) is the additive identity of the ring.

First, find the additive identity of \(R\). Let \((z_1, z_2)\) be the additive identity. Then \((a, b) + (z_1, z_2) = (a, b)\) for all \((a, b) \in R\).

Using the addition rule: \((a + z_1, b + z_2) = (a, b)\).

This implies \(a + z_1 = a\) (mod 2) and \(b + z_2 = b\) (mod 2). In \(Z_2\), the additive identity is 0. So, \(z_1 = 0\) and \(z_2 = 0\).

The additive identity (zero element) of \(R\) is \((0, 0)\).

Now, we look for non-zero elements \((a, b)\) and \((c, d)\) in \(R\) such that \((a, b) \cdot (c, d) = (0, 0)\).

We need \((a \cdot c, b \cdot d) = (0, 0)\).

This requires \(a \cdot c = 0\) (mod 2) and \(b \cdot d = 0\) (mod 2). In \(Z_2\), the product is 0 only if at least one of the factors is 0.

Let's consider non-zero elements in \(R\). Some non-zero elements are \((1, 0)\), \((0, 1)\), \((1, 1)\).

Consider \((1, 0)\). This is a non-zero element. Is there a non-zero element \((c, d)\) such that \((1, 0) \cdot (c, d) = (0, 0)\)?

\((1 \cdot c, 0 \cdot d) = (0, 0)\) which means \((c, 0) = (0, 0)\).

This implies \(c = 0\). The element is \((0, d)\). If we choose \(d=1\), we get the element \((0, 1)\). \((0, 1)\) is a non-zero element.

Let's check the product: \((1, 0) \cdot (0, 1) = (1 \cdot 0, 0 \cdot 1) = (0, 0)\).

Since \((1, 0) \neq (0, 0)\) and \((0, 1) \neq (0, 0)\), and their product is \((0, 0)\), both \((1, 0)\) and \((0, 1)\) are zero divisors in \(R\).

Therefore, \(R\) contains divisors of zero.

Summary of Ring Properties

Based on our analysis, the ring \(R = (Z_2 \times Z_2, +,.)\) has the following properties:

• It is a commutative ring.
• It is a ring with unity, the unity element being \((1, 1)\).
• It contains divisors of zero, for example, \((1, 0)\) and \((0, 1)\).

Comparing these properties with the given options:

• Option 1: R is a commutative ring with unity and it contains divisor of zero. (Matches our findings)
• Option 2: R is a non commutative ring with unity and it contains divisor of zero. (Incorrect)
• Option 3: R is a commutative ring without unity and it contains no divisor of zero. (Incorrect)
• Option 4: R is a commutative ring with unity and it contains no divisor of zero. (Incorrect)

The properties derived for the ring \(R\) match the description in the first option.