dinvertibilité

Dinvertibility is a concept in abstract algebra, specifically within the study of rings and modules. It describes a property of elements within a ring that are related to their multiplicative inverse. An element 'a' in a ring R is considered dinvertible if there exists an element 'b' in R such that ab = 1 and ba = 1, where '1' is the multiplicative identity of the ring. This 'b' is then called the dinverse of 'a'. The term "dinvertible" is often used to distinguish from elements that may have a right inverse (only ab=1) or a left inverse (only ba=1), but not necessarily both. In rings where multiplication is commutative, the distinction between a right inverse, a left inverse, and a two-sided inverse (dinverse) collapses, and such elements are simply referred to as invertible or units. The set of all dinvertible elements in a ring forms a group under multiplication, known as the group of units. Studying dinvertibility is crucial for understanding the structure of rings and the behavior of their elements, particularly in areas like field theory and linear algebra where matrices, which form a non-commutative ring, are frequently encountered.