We discuss in more detail about the Diamond Lemma in rings, following The Diamond Lemma for Ring Theory by Bergman.
Let \(X\) be a set, \(\langle X \rangle\) the free semigroup, \(k\langle X\rangle\) the free associative algebra.
\(S\) be the set \(\{(w_i, f_i)\}\) of pairs of leading terms \(w_i\) and relations \(f_i\), for \(A, B\) in \(\langle X\rangle\), the map \(r_{A\sigma_i B}\) is the \(k\)-linear reduction mapping of \(\mathrm{End}(k\langle X\rangle)\) that maps \(Aw_iB \mapsto Af_iB\), and fixes all other basis elements.
An element \(a\in k\langle X\rangle\) is reduction-finite if for every infinite sequence of reductions, there is a point from which the rest of the reductions act trivially.
reduction-unique if it is reduction finite and all final sequences of reductions give the same result which we denote by \(r_S(a)\).
The set of reduction-finite elements form a submodule. This is obvious since the reduction is linear and has to stop on each component.
An element \(a\) is reduction-unique, iff starting from any finite sequence of reductions \(r\), \(ra\) is reduction-unique.
Reduction-finite means there exists a final sequence. The final result might not be unique though.
The set of reduction-unique elements is a \(k\)-submodule of \(k\langle X\rangle\), and \(r_S\) can be defined on this submodule, into the set of irreducible elements (elements on which all reductions act trivially).
\[ r_S : \mathrm{RedUni}(k\langle X\rangle) \to \mathrm{Irred}(k\langle X\rangle) \]
Because starting from any sequence we can choose a final sequence \(r\) of \(a+b\), from which we can find \(r'r(a)\) final and \(r''r'r(b)\) final, so \(r''r'r(a+b) = r''r'r(a) + r''r'r(b)\).
Let \(a,b,c\in k\langle X\rangle\) and assume that for all monomials \(A,B,C\) inside \(a,b,c\) respectively, the product \(ABC\) is reduction-unique, then \(abc\) is reduction-unique and this is true for all \(ar(b)c\), for which they reach the same results \(r_S(ar(b)c) = r_S(abc)\).
For reductions \(\alpha, \beta\) and words \(A,B,C\) that \(w_\alpha = AB\), \(w_\beta=BC\), we say that this is an overlap ambiguity. It is resolvable if exists sequence \(r\) such that they reduce \(ABC\) to the same result,
\[r\alpha(ABC) = r(f_\alpha C) = r\beta(ABC) = r(Af_\beta).\]
Similarly if you want to reduce \(ABC\) where you can also reduce \(B\), we have an inclusion ambiguity. It is resolvable if there exists a sequence \(r\) such that \(r\alpha(ABC) = r(Af_\alpha C) = r\beta(ABC) = r(g_\beta)\).
A semigroup partial order \(\le\) is a partial order that is compatible with multiplications.
We say it is compatible with reduction system \(S = \{(w_i,f_i)\}\) if for all \(\sigma\in S\), \(\mathrm{Terms}(f_\sigma) < w_\sigma\). i.e. reduction always strictly reduces the leading term, or equivalently say that \(w_\sigma\) is the leading term of \(f_\sigma\).
Let \(I\) be the ideal of \(k\langle X\rangle\) generated by relations \(w_i-f_i\), i.e. a \(k\)-module generated by \(A(w_i-f_i)B\). As an approximation we can form the submodule (not ideal) \(I_A\) denoting the submodule of \(k\langle X\rangle\) spanned by elements of the form \(B(w_i-f_i)C\) such that \(Bw_iC < A\). These are the relations whose leading term \(<A\).
The Diamond Lemma states that given a semigroup partial order (partial orders that are compatible with multiplications) \(\le\) and compatible with the reduction system \(S\),
The following are equivalent:
All ambiguities of \(S\) are resolvable.
All ambiguities of \(S\) are resolvable relative to \(\le\).
All elements of \(k\langle X\rangle\) are reduction-unique under \(S\).
A set of representatives in \(k\langle X\rangle\) for the elements of \(R=k\langle X\rangle/I\) is given by \(\mathrm{Irred}(k\langle X\rangle)\), spanned by \(S\)-irreducible elements.