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 Nov2 awarded Teacher Nov1 answered A ruler with missing graduation. May10 comment What do ideals of a ring say about its inner structure? Could you give an example of a property of a ring not following from its ideals but only from its modules? May7 comment What do ideals of a ring say about its inner structure? If there's a non-trivial ideal $I$ in a ring, then for any $x\in I$, $x$ does not have an inverse? Otherwise, $I$ would cover the whole ring. May7 awarded Supporter May7 asked What do ideals of a ring say about its inner structure? Oct26 comment From a deterministic discrete process to a Markov chain: conditions? I was thinking about the explanation how to obtain a probabilistic process from a deterministic process (3rd paragraph in your post). As you said, given a deterministic process $a_t=F(a_{t-1})$, we make it probabilistic by making $a_0$ a random variable. If $a_0$ is fixed, we get only a degenerate chain with prob's 0/1. However, how about keep $a_0$ fixed, consider the chain homogenous and let the transition probabilities be the average over many deterministic transitions? (e.g., if the deterministic chain goes from $s_0$ to $s_1$ 1/3 times over many steps, $p(s_1|s_0):=\frac{1}{3}$) Oct25 awarded Editor Oct25 revised From a deterministic discrete process to a Markov chain: conditions? added 648 characters in body Oct25 comment From a deterministic discrete process to a Markov chain: conditions? However, my question remains: in what case would this yield a M. process (under the assumption that $a_0$ follows some starting distribution)? You said that its's not likely to produce a Markov process, but is there a Theorem that confirms this? Otherwise, there must be some conditions on the "abstraction" function and the process? I'm looking for some references or book? Oct25 comment From a deterministic discrete process to a Markov chain: conditions? @Dider: thanks! As you and Craig pointed out example #1 was not good (gives a degenerate process which is not interesting). As you said, what I had in mind is that $a_0$ follows some starting distribution and then we obtain a random process. Oct25 comment From a deterministic discrete process to a Markov chain: conditions? Thanks, but I don't think this answers the question. It wasn't about whether the deterministic process is a Markov chain or not (it was about whether the resulting probabilistic process is a Markov chain or not). Oct25 awarded Student Oct25 asked From a deterministic discrete process to a Markov chain: conditions?