- #1
mathmari
Gold Member
MHB
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Hey! ![Eek! :eek: :eek:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
I want to check if the following statement are true.
Let $R$ be a ring, $S$ a subring and $I$ an ideal.
I want to check if the following statement are true.
Let $R$ be a ring, $S$ a subring and $I$ an ideal.
- If $R$ is Noetherian then $S$ is also.
- If $R$ is Noetherian then $R/I$ is also.
- If $R$ is Artinian then $S$ is also.
- If $R$ is Artinian then $R/I$ is also.
- If $R$ is P.I.D. then $S$ is also.
- If $R$ is P.I.D. then $R/I$ is also.
- If $R$ is U.F.D. then $S$ is also.
- If $R$ is U.F.D. then $R/I$ is also.
- If $R$ is an euclidean domain then $S$ is also.
- $R$ is Noetherian iff each increasing sequence of ideal $I_1\subseteq I_2 \subseteq I_3 \subseteq \dots \subseteq I_k\subseteq \dots $ stops, i.e., $\exists k$ such that $I_k=I_{k+1}$, right?
Then since $S$ is a subring of $R$, not all $I_i$ are contained in $S$. Therefore, the above condition isn't necessarily satisfied. So, $S$ is not necessarily Noetherian.
is this correct? (Wondering) - What can we say in that case? Does the increasing sequence stop? (Wondering)
- $R$ is Artinian iff each decreasing sequence of ideal $I_1\supseteq I_2 \supseteq I_3 \supseteq \dots \supseteq I_k\supseteq \dots $ stops, i.e., $\exists k$ such that $I_k=I_{k+1}$, right?
Then since $S$ is a subring of $R$, not all $I_i$ are contained in $S$. Therefore, the above condition isn't necessarily satisfied. So, $S$ is not necessarily Artinian.
is this correct? (Wondering) - What can we say in that case? Does the decreasing sequence stop? (Wondering)
- If $R$ is P.I.D. then the ideals are prime, therefore $S$ contain also only prime ideals. So, $S$ is also P.I.D., right? (Wondering)
- What can we say in this case? (Wondering)
- If $R$ is U.F.D. then $\forall r\in R\setminus \{0\}$, $r\notin U(R)$: $r=a_1 \cdots a_k$ with $a_i$ irreducible, and if $r=a_1\cdots a_k=b_1\cdots b_t$ with $a_i, b_i$ irreducible then $k=t$ and $a_i=b_iu_i$ with $u_i\in U(R), \forall i=1, \dots , k$.
Does the same hold also for $S$ ? (Wondering) - And also in this case? (Wondering)
- How can we check that? (Wondering)