Ideals of $\mathbb{Z}_6\times\mathbb{Z}_{10}$ Two things concerning the ring $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$:


*

*Identify a subset of $R$ that is a subring of $R$ but not an ideal of $R$.

*Identify a maximal ideal in $R$.


For the first part, how about $\{(0,0), (3,5)\}$? It is a subring but $(3,5)(2,5)=(0,5)$ so it is not an ideal. 
Secondly, how about $\{(a,b)\mid a\in\mathbb{Z}_6\land b\in\mathbb{Z}_{10}\land a,b\in O\}$, where $O$ is the set of even integers. Is this a maximal ideal of $R$?
Can anyone correct me where I'm going wrong here? 

*

*If you form a subset by repeatedly adding $(1,1)$ to itself then you get a subring of cardinality $\text{lcm}(6,10) = 30$. But $(1,1)$ is the unity in $R$ so the only ideal which contains $(1,1)$ is the whole ring. So this subring is not an ideal. Also, turns out your example is correct. A subring must contain the unity of the containing ring, so your example is not a subring.

*Your example is correct.


It looks like you understand this.
