A quotient ring is a ring that is formed by dividing a larger ring by a smaller subring. The elements of the quotient ring are the elements of the larger ri...
A quotient ring is a ring that is formed by dividing a larger ring by a smaller subring. The elements of the quotient ring are the elements of the larger ring that are left unchanged when the larger ring is divided by the smaller subring.
The construction of a quotient ring is based on the following observation:
The set of all elements in the larger ring that are not in the subring is naturally equipped with a natural structure of a ring.
This ring is called the quotient ring of the larger ring and is denoted by the symbol R/S.
The elements of the quotient ring are classified into two types: those elements that are in the subring and those that are not.
Elements in the subring are exactly those elements of the quotient ring that are not in the subring.
Elements not in the subring are precisely those elements of the quotient ring that are in the subring.
The kernel of the quotient ring is the subset of all elements of the larger ring that are in the subring. It is denoted by Ker(R/S).
The image of an element in the larger ring under the quotient map is the element in the quotient ring corresponding to that element in the larger ring.
The ideal of the quotient ring is the subset of all elements of the larger ring that are mapped to the zero element under the quotient map. It is denoted by I(R/S).
The quotient ring inherits the structure of the larger ring, including the addition, subtraction, multiplication, and division of elements, as well as the concept of ideals. The ideal of the quotient ring is the largest ideal of the larger ring that is contained in the subring.
The quotient ring is a very important object in ring theory and linear algebra. It has a wide range of applications in mathematics, including the study of homomorphisms, invariant theory, and representation theory