I am very new at algebra and currently studying at ring. Now I can find the explanation of "Subring is not necessarily ideal" from
but I still do not understand. Can someone explain it to me? Thanks!
2026-03-28 20:54:01.1774731241
Subring is not necessarily ideal, where to start?
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Maybe it helps to clarify the definitions. So a subring of a ring $R$ is a subset $S$ of $R$ preserving the structure of the ring (see for example https://en.wikipedia.org/wiki/Subring).
Now for an ideal, say $I$ of a ring $R$ we have the property, that $$\forall x \in I ~\forall r \in R ~~ x \cdot r \in I$$ (see for instance https://en.wikipedia.org/wiki/Ideal_(ring_theory) for a definition of ideal).
So a subring does not have to have this property (by definition), which is used in the proof https://proofwiki.org/wiki/Subring_is_not_necessarily_Ideal.