Definition 1.3.1. (Zero-divisors).
A nonzero element \(r\) of a ring \(R\) is said to be a left \(0\)-divisor if there exists a nonzero element \(s\in R\) such that \(rs=0\text{.}\)
Right \(0\)-divisors are defined similarly.
Section 1.3 Types of ring elements
Remark 1.3.2.
A left \(0\)-divisor need not be a right \(0\)-divisor. For instance, consider a triangular ring given in Example 1.2.7 with \(A=\Z/2\Z\text{.}\) We denote the additive identity in the ring \(\Z/2\Z\) by \([0]\text{.}\) Verify that the following element is a left \(0\)-divisor but not a right \(0\)-divisor.
\begin{equation*}
\begin{pmatrix}2\amp [0]\\0_{\Z}\amp 1\end{pmatrix}
\end{equation*}
Definition 1.3.3. (Nilpotent Element).
An element \(r\) of a ring \(R\) said to be nilpotent if there exists a natural number \(n\) such that \(r^n=0\) in \(R\text{.}\)Definition 1.3.4. (Idempotent Elements and Orthogonal Idempotents).
An element \(r\) of a ring \(R\) said to be idempotent if \(r^2=r\) in \(R\text{.}\)
Idempotents \(0,1\) of a ring \(R\) are called the trivial idempotents.
Idempotents \(\{e_i\}_{i\in I}\) are called orthogonal if \(e_ie_j=0=e_je_i\) for every \(i\neq j\text{.}\)
Definition 1.3.5. (Left and right invertible elements).
Let \(R\) be a ring. A nonzero element \(r\in R\) is said to be left invertible (resp., right invertible) if there exists \(s\in R\) (resp., \(s^\prime\in R\)) such that \(sr=1\) (resp., \(rs^\prime=1\)), and we call \(s\) (resp., \(s^\prime\)) to be a left inverse (resp., a right inverse) of \(r\text{.}\)Lemma 1.3.6.
Let \(R\) be a ring, and \(0\neq r\in R\text{.}\) If \(s\) is a left inverse of \(r\) and \(t\) is a right inverse of \(r\) then \(s=t\text{.}\)Proof.
Convention 1.3.7.
Suppose that \(0\neq r\in R\) has both left and right inverse, say \(s\) and \(t\text{,}\) respectively. We denote \(s=t=r^{-1}\) and \(r^{-1}\) is called the inverse of \(r\text{.}\) Thus, \(r^{-1}r=1=rr^{-1}\text{.}\) The use of an article ‘the’ before the term inverse is justified in the following Lemma 1.3.8.
