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Let $X$ be an alphabet, $u,v,p,q,r,s$ be words in the alphabet $X$. I am looking for four elements in the free associative ring $R$ (i.e. four linear combinations of words in $X$) $x,y,z,t$ such that $$u-v=x(p-q)y+z(r-s)t.$$ Is this problem decidable?

The problem is motivated by the need to define an analog of Dehn functions for associative rings. In groups, the Dehn function is recursive iff the word problem is decidable. The question (asked by E. Zelmanov) is whether the same is true for rings.

Update: An easier problem: is there an algorithm for solving the equation

$$u-v=x(p-q)y$$ (one summand)?

1

# An equation in the free associative ring

Let $X$ be an alphabet, $u,v,p,q,r,s$ be words in the alphabet $X$. I am looking for four elements in the free associative ring $R$ (i.e. four linear combinations of words in $X$) $x,y,z,t$ such that $$u-v=x(p-q)y+z(r-s)t.$$ Is this problem decidable?

The problem is motivated by the need to define an analog of Dehn functions for associative rings. In groups, the Dehn function is recursive iff the word problem is decidable. The question (asked by E. Zelmanov) is whether the same is true for rings.