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September 19

Does anyone here know the longest distance to zero found by now?

In other words, what's the lowest number the 50 move rule would have to be to not turn a mate to a draw with perfect play? (as far as we know, 10,000 terabytes would be needed just to store the best move for each position with ≤6 men and 2 kings) Sagittarian Milky Way (talk) 13:09, 19 September 2018 (UTC)[reply]

I think you're asking what chess position requires the largest number of moves to result in mate, without any captures or pawn moves. The article that you linked to yourself says that there is a position that requires 545 moves, found in 2013 (citation here). Is that not what you're asking? CodeTalker (talk) 23:17, 19 September 2018 (UTC)[reply]

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

That's not 545, the first capture is on move 514 and as that database optimizes for distance to mate that isn't the longest one in the database anyway. I've found that [1] has released the first 7-man DTZ database 32 days ago (6 years after the first 7-man distance to mate database and 13 years after the first 6-man database). That site explains the difference between DTZ and DTM and shows that this position with black to move is the longest (517 moves of maneuvering before white can eat anything with perfect play (if the 50- and 75-move rules didn't exist) Then white mates in a flash compared to 517 moves). Sagittarian Milky Way (talk) 05:20, 20 September 2018 (UTC)[reply]

Non-isomorphic abelian groups with isomorphic endomorphism rings

Can two non-isomorphic abelian groups A and B have isomorphic endomorphism rings? If so, for which rings R does A (or equivalently, B) have a left R-module structure? GeoffreyT2000 (talk) 16:06, 19 September 2018 (UTC)[reply]

According to [2], yes, i.e. the endomorphism ring does not determine the group. The example they give is End(Z)≅Z≅End(A) where A is the set of fractions with square-free denominators. There's no proof but it looks straightforward. Not sure about the second half of your question, but for A=Z it seems likely that the answer is well known. --RDBury (talk) 07:30, 20 September 2018 (UTC)[reply]

For a ring R and an abelian group A, giving a left R-module structure on A is the same thing as giving a ring-homomorphism

R\to End(A)

--176.230.125.208 (talk) 15:03, 24 September 2018 (UTC)[reply]

To be precise, for a ring R, the following are equivalent:

Z has a left R-module structure.
Z has a right R-module structure.
R has a ring homomorphism into Z (necessarily surjective).
R has a two-sided ideal I for which the quotient ring R/I is isomorphic to Z.
R is isomorphic to the free unital ring on some rng, which may or may not have its own unit element.

GeoffreyT2000 (talk) 00:23, 25 September 2018 (UTC)[reply]