Suppose that $R$ is a commutative ring and $|R|=30$. If $I$ is an ideal of $R$ and $|I|=10$, prove that $I$ is maximal ideal

Solution: $|R/I|=3 \implies R/I \approx Z_3$ which is a field.

If $R$ is a commutative ring with unity and $I$ is an ideal, then $R/I$ is a field if and only if $I$ is a maximal ideal. Hence, in this problem, $I$ is a maximal ideal iff $R$ contains a unity.

How do we prove that $R$ must contain a unity?

I know that a finite commutative ring with no zero divisors definitely contains a unity. But, then $R$ has not been stated to not contain zero divisors either.

Thank you for your help..

  • $\begingroup$ Where is the problem from? Are you sure that rings are not assumed to have an identity element? $\endgroup$ – Jonas Meyer Jul 8 '14 at 5:09
  • $\begingroup$ It's from Gallian. In Gallian, ring need not necessarily have a unity element $\endgroup$ – MathMan Jul 8 '14 at 5:09
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    $\begingroup$ Even with the zero product, a ring with 3 elements is simple. $\endgroup$ – Jonas Meyer Jul 8 '14 at 5:12
  • $\begingroup$ uhmm, .. $R/I = \{I,a+I,2a+I\}, a \notin I, 3a=0$. Why does that $I$ maximal? $\endgroup$ – MathMan Jul 8 '14 at 5:19
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    $\begingroup$ If $|R/I|=3$, then $R/I$ is simple because it has only $\{0\}$ and itself as additive subgroups. In general, $R/I$ is simple if and only if $I$ is maximal. $\endgroup$ – Jonas Meyer Jul 8 '14 at 5:21

Let $x \in R-I$. Consider the ideal $J=\langle I, x \rangle$. Observe that $|J|>10$. Moreover this must be a subgroup of $R$ but based on Lagrange its order should divide $30$. But $I \leq J$ as well, therefore $10$ divides $|J|$. Thus $J=R$, hence $I$ is maximal.

  • $\begingroup$ what does $\langle I,x\rangle$ means? $\endgroup$ – Anish Ray Aug 23 at 16:58
  • $\begingroup$ @AnishRay It is the ideal generated by the set (ideal) $I$ and element $x$. If the ring is commutative and has unity (some assume unity as part of the definition and some don't) then $$\langle I,x \rangle=\{j+rx \, | \, r \in R, j \in I\}.$$ $\endgroup$ – Anurag A Aug 23 at 18:50

The ideal $I$ is maximal because $R/I$ is simple.


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