Could you explain how to show 8.12 in detail? I also attached definitions and theorems in my textbook. **Theorem 8.2.** Let \( R \) be a ring. Then the additive identity, 0, is unique. If \( R \) has a multiplicative identity 1, then it too is unique.**Theorem 8.3.** Let \( R \) be a ring. If \( a, b \in R \), then1. \( 0a = a0 = 0 \);2. \((-a)b = a(-b) = -(ab)\); and3. \((-a)(-b) = ab\).**Corollary 8.1.** If \( R \) is a ring with identity, then \((-1)a = -a\), for any \( a \in R \).**Theorem 8.4.** Let \( R \) be any ring, and \( a_1, a_2, \ldots, a_n \in R \). Then regardless of how the product \( a_1a_2 \cdots a_n \) is bracketed, the result equals \( \cdots (((a_1a_2)a_3)a_4) \cdots a_{n-1})a_n \).**Example 8.6.** In \( \mathbb{Z}_6 \), we have \( 2 \cdot 3 = 0 \), but \( 2 \neq 0 \) and \( 3 \neq 0 \).**Example 8.7.** In \( M_2(\mathbb{R}) \), we have\[\begin{pmatrix}1 & 2 \\3 & 6 \end{pmatrix}\begin{pmatrix}-2 & 4 \\1 & -2 \end{pmatrix}=\begin{pmatrix}0 & 0 \\0 & 0 \end{pmatrix},\]but \[\begin{pmatrix}1 & 2 \\3 & 6 \end{pmatrix}\neq\begin{pmatrix}0 & 0 \\0 & 0 \end{pmatrix}\neq\begin{pmatrix}-2 & 4 \\1 & -2 \end{pmatrix}.\]In dealing with groups, **8.12.** Let \( R \) be a ring with identity. Suppose that there exist \( a, b, c \in R \) such that \( ab = ba = 1 \) and \( ac = 0 \). Show that \( c = 0 \).**8.16.** Let \( R \) be a ring in which \( a^2 = a \) for every \( a \in R \).1. Show that \( a + a = 0 \) for every \( a \in R \).2. Show that \( R \) is commutative.

Given that R is a ring with identity. The multiplicative identity of a ring is denoted by 1 and the additive identity is denoted by 0. Then, by the property of a ring, for any a∈R, we have a1=1a=a and a0=0a=0. Further, the given ring has the property that there exists a, b, c∈R such that ab=ba=1 and ac=0. We have to prove that c=0.We have, c=1c ∵c1=1c=c=bac ∵ab=ba=1=bac ∵being a ring, R is associative under multiplication=b0 ∵ac=0=0 ∵b0=0b=0 Thus, we have proved that c=0.