Difference between revisions of "Subring"

Latest revision as of 15:37, 16 June 2008

Given a ring $R$ , a subset $Q \subset R$ is called a subring of $R$ if it inherits the ring structure from $R$ . That is, $Q$ must contain both the $0$ and $1$ (additive and multiplicative identities) of $R$ and be closed under the ring operations of multiplication, addition and additive inverse-taking.

Examples

Consider the ring $R = \mathbb{Z} \times \mathbb{Z}$ of ordered pairs of integers with coordinatewise operations, i.e. $(a, b) + (c, d) = (a + c, b + d)$ and $(a, b) \cdot (c, d) = (ac, bd)$ . Then the diagonal ring $D = \{(a, a) \mid a \in \mathbb{Z}\}$ is a subring of $R$ : it contains the additive identity $(0, 0)$ , the multiplicative identity $(1, 1)$ and is closed under multiplication and addition.

Non-examples

The notion of a subring is slightly more subtle than that of a subgroup. Suppose that $R$ is a commutative ring with an idempotent element $i$ other than $0$ and $1$ , i.e. $i$ is a solution to the equation $i^2 = i$ . Consider the principle ideal $I = Ri = \{a \in R \mid \exists b, a = bi\}$ . As an ideal, this set is closed under addition and multiplication and contains the additive identity of $R$ . Moreover, this ideal is a ring with multiplicative identity $i$ : $i \cdot bi = bi^2 = bi$ for every $b \in R$ , so $i\cdot a = a$ for every $a \in I$ . However, it is not a subring of $R$ because it does not contain the multiplicative identity of $R$ . (Otherwise $1 \in I$ and there is some $j \in R$ such that $ij = 1$ , so $i^2j = i$ but also $i^2j = ij = 1$ , and we assumed $i \neq 1$ , a contradiction.)

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-Let <math>R</math> be a [[ring]].  A subset <math>Q</math> of <math>R</math> is called a '''subring''' of <math>R</math> if it has an induced ring structure.
+Given a [[ring]] <math>R</math>, a [[subset]] <math>Q \subset R</math> is called a '''subring''' of <math>R</math> if it inherits the ring structure from <math>R</math>.  That is, <math>Q</math> must contain both the <math>0</math> and <math>1</math> (additive and multiplicative [[identity | identities]]) of <math>R</math> and be [[closed]] under the ring [[operation]]s of multiplication, addition and additive inverse-taking.
+== Examples ==
+Consider the ring <math>R = \mathbb{Z} \times \mathbb{Z}</math> of [[ordered pair]]s of [[integer]]s with coordinatewise operations, i.e. <math>(a, b) + (c, d) = (a + c, b + d)</math> and <math>(a, b) \cdot (c, d) = (ac, bd)</math>.  Then the diagonal ring <math>D = \{(a, a) \mid a \in \mathbb{Z}\}</math> is a subring of <math>R</math>: it contains the additive identity <math>(0, 0)</math>, the multiplicative identity <math>(1, 1)</math> and is closed under multiplication and addition.
+== Non-examples ==
+The notion of a subring is slightly more subtle than that of a [[subgroup]].  Suppose that <math>R</math> is a [[commutative ring]] with an [[idempotent]] [[element]] <math>i</math> other than <math>0</math> and <math>1</math>, i.e. <math>i</math> is a solution to the equation <math>i^2 = i</math>.  Consider the [[principle ideal]] <math>I = Ri = \{a \in R \mid \exists b, a = bi\}</math>.  As an [[ideal]], this set is closed under addition and multiplication and contains the additive identity of <math>R</math>.  Moreover, this ideal is a ring with multiplicative identity <math>i</math>: <math>i \cdot bi = bi^2 = bi</math> for every <math>b \in R</math>, so <math>i\cdot a = a</math> for every <math>a \in I</math>.  However, it is ''not'' a subring of <math>R</math> because it does not contain the multiplicative identity of <math>R</math>.  (Otherwise <math>1 \in I</math> and there is some <math>j \in R</math> such that <math>ij = 1</math>, so <math>i^2j = i</math> but also <math>i^2j = ij  = 1</math>, and we assumed <math>i \neq 1</math>, a contradiction.)
-{{stub}}
 == See also ==
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 [[Category:Ring theory]]
+{{stub}}

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Difference between revisions of "Subring"

Latest revision as of 15:37, 16 June 2008

Examples

Non-examples

See also