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**Reversible rings.**
*(English)*
Zbl 1021.16019

A ring \(R\) is called reversible if \(ab=0\) implies \(ba=0\); it is insertable if \(ab=0\) implies \(aRb=0\). Clearly any reversible ring is insertable, but not conversely. The author proves that a ring is (i) an integral domain if and only if it is prime and reversible (insertable), (ii) reduced if and only if it is semiprime and reversible (insertable). He also answers Köthe’s conjecture for reversible rings by proving that in any reversible ring the set of all nilpotent elements is a nil ideal. More generally, a ring \(R\) is called fully reversible if for any two square matrices \(A\), \(B\) of the same order, \(AB\) is full if and only if \(BA\) is full [cf. P. M. Cohn, Free rings and their relations, Lond. Math. Soc. Monogr. No. 19, Academic Press (1985; Zbl 0659.16001) for unexplained terms]; likewise, \(R\) is fully insertable if \(AB\) non-full implies \(AXB\) non-full for any matrix \(X\) of the same order; if this holds only for invertible \(X\), the ring is called unit-stable.

Now the main theorem states: For any non-zero ring \(R\) the following conditions are equivalent: (a) \(R\) is fully reversible, (b) \(R\) is fully insertable and every full matrix is stably full, (c) \(R\) is unit-stable and every full matrix is stably full, (d) the least matrix ideal of \(R\) is proper and consists entirely of non-full matrices. The proof uses the magic lemma (Lemma 7.9.1, loc. cit. p. 446) and the fact that over a fully reversible ring every full matrix is stably full. Applications include the result that every fully reversible semiprime ring is a subring of a direct product of skew fields and a fully reversible ring is embeddable in a skew field if and only if it is an integral domain or equivalently, a prime ring.

Now the main theorem states: For any non-zero ring \(R\) the following conditions are equivalent: (a) \(R\) is fully reversible, (b) \(R\) is fully insertable and every full matrix is stably full, (c) \(R\) is unit-stable and every full matrix is stably full, (d) the least matrix ideal of \(R\) is proper and consists entirely of non-full matrices. The proof uses the magic lemma (Lemma 7.9.1, loc. cit. p. 446) and the fact that over a fully reversible ring every full matrix is stably full. Applications include the result that every fully reversible semiprime ring is a subring of a direct product of skew fields and a fully reversible ring is embeddable in a skew field if and only if it is an integral domain or equivalently, a prime ring.

Reviewer: Paul M.Cohn (London)

### MSC:

16U30 | Divisibility, noncommutative UFDs |

16U80 | Generalizations of commutativity (associative rings and algebras) |

16S50 | Endomorphism rings; matrix rings |

16K40 | Infinite-dimensional and general division rings |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16N60 | Prime and semiprime associative rings |