In this section we define notions and terminology related to subrings and ideals of a ring. In the next section we give several examples related to these notions.
Definition1.5.1.(Subring).
A subset \(S\) of a ring \(R\) is said to be a subring of \(R\) if it satisfy the following conditions.
The addition inherited from \(R\) makes \(S\) a subgroup of \((R,+)\text{.}\)
The unity \(1\) of \(R\) belongs to \(S\text{.}\)
\(S\) is closed under multiplication inherited from \(R\text{,}\) i.e., for any \(s,t\in S\subseteq R\) we have \(st\in S\text{.}\)
Remark1.5.2.
A subring \(S\) of a ring \(R\) is itself is a ring with unity. Furthermore, the unity of \(S\) is the same as the unity \(R\text{.}\)
Lemma1.5.3.
Suppose that \(S_i\) (\(i\in I\)) are subrings of a ring \(R\text{.}\) The intersection of \(S_i\text{,}\)\(\bigcap_iS_i\) is a subring of \(R\text{.}\)
Since the intersection of subgroups is a subgroup, \(\bigcap_iS_i\) is a subgroup of \(R\) under addition. Since \(1\in S_i\) for every \(i\) we get that \(1\in\bigcap_iS_i\text{.}\) We now show that \(\bigcap_iS_i\) is closed under multiplication. If \(x,y\in\bigcap_iS_i\) then \(xy\in S_i\) for every \(i\) because \(S_i\) is a subring. Consequently, \(xy\in\bigcap_iS_i\text{.}\) Hence \(\bigcap_iS_i\) is a subring.
Definition1.5.4.(Subring generated by a subset).
A subring generated by a nonempty subset \(X\) of a ring \(R\) is the intersection of all subrings of \(R\) which contain \(X\text{.}\)
Definition1.5.5.(Left Ideal and Right Ideal).
Let \(R\) be a ring. A nonempty subset \(I\) of \(R\) is said to be a left ideal (resp., a right ideal) of \(R\) if \((I,+)\) is a subgroup of \((R,+)\text{,}\) and for every \(r\in R\) and every \(x\in I\) we have \(rx\in I\) (resp., \(xr\in I\)).
Definition1.5.6.(Ideal).
A nonempty subset \(I\) of a ring \(R\) is said to be an ideal of \(R\) or a two-sided ideal of \(R\) if \(I\) is both a left and a right ideal of \(R\text{.}\) Thus, \(I\) is an ideal if \((I,+)\) is a subgroup of \((R,+)\) and for every \(r,s\in R\) and every \(x\in I\) we have \(rxs\in I\text{.}\)
Definition1.5.7.(Proper ideal).
A left ideal (resp. right ideal, two-sided ideal) \(I\) in a ring \(R\) is said to be a proper ideal if \(I\neq R\text{.}\)
Definition1.5.8.(Maximal ideal).
Let \(R\) be a ring. A left ideal (resp., right ideal or two-sided ideal) of \(R\) is said to be a maximal left ideal (resp., maximal right ideal or maximal ideal) of \(R\) if it is maximal element of the set of all left ideals (resp., right ideals or two-sided ideals) distinct from \(R\text{.}\)
In other words, a left ideal (resp., right ideal or two-sided ideal) \(I\neq R\) is maximal if only left ideal (resp., right ideal or two-sided ideal) of \(R\) containing \(I\) are \(I\) and \(R\text{.}\)
Definition1.5.9.(Prime ideal in a commutative ring).
Let \(A\) be a commutative ring. A proper ideal \(I\) in \(A\) is said to be a prime ideal if \(ab\in I\) then either \(a\in I\) or \(b\in I\text{.}\)
Definition1.5.10.(Simple Rings).
A ring \(R\) is said to be simple if only ideals, i.e., only two-sided ideals of \(R\) are \(\{0\}\) and \(R\) only.
Definition1.5.11.(Subalgebra).
Let \(F\) be a field and let \(A\) be an \(F\)-algebra. A nonempty subset \(B\) of \(A\) is said to be an \(F\)-subalgebra of \(A\) if \(B\) is a subring of \(A\) and also a vector subspace of \(A\text{.}\)
Definition1.5.12.(Subalgebra generated by a subset).
Let \(A\) be an \(F\)-algebra and \(S\) be a nonempty subset of \(A\text{.}\) The subalgebra generated by \(S\) is the intersection of all subalgebras of \(A\) containing \(S\text{.}\) It is denoted by \(F[S]\text{.}\) Thus,
\begin{align*}
F[S]\amp=\bigg\{a_0\cdot 1+\sum a_{i_1\cdots i_r}\cdot s_{i_1}s_{i_2}\cdots s_{i_r}:\;i_j,r\in\N\text{ and }s_{i_j}\in S\\
\amp\quad\qquad\qquad a_0,a_{i_1\cdots i_r}\in F\text{ and all but finitely many }a_{i_1\ldots i_r}= 0\bigg\}
\end{align*}
Definition1.5.13.(Ideal in an algebra).
A nonempty subset \(I\) of an \(F\)-algebra \(A\) is said to be an ideal in \(A\) if \(I\) is an ideal of \(A\) as ring and a subspace of \(A\) as a vector space over \(F\text{.}\)
