QUOTIENT RING Meaning and
Definition
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A quotient ring, in the field of abstract algebra, is a mathematical concept used in ring theory. A ring is an algebraic structure consisting of a set of elements equipped with two binary operations, typically addition and multiplication, which satisfy certain axioms. The quotient ring is a construction obtained by partitioning the set of elements of a given ring into equivalence classes, and then defining addition and multiplication operations on these equivalence classes.
Formally, let R be a ring and I be an ideal of R. An ideal is a subset of a ring that is closed under addition, subtraction, and multiplication by elements of the ring. The quotient ring R/I is then defined as the set of equivalence classes of R modulo I, where two elements a and b in R belong to the same equivalence class if and only if their difference a - b lies in I.
The addition operation in the quotient ring is defined by adding representatives of the equivalence classes, while the multiplication operation is defined by multiplying representatives of the equivalence classes. These operations are well-defined and satisfy all the ring axioms, making the quotient ring itself a ring.
Quotient rings are widely used in mathematics, particularly in the study of abstract algebra, algebraic geometry, and number theory. They provide a useful framework for studying ring structures in a more tractable and understandable manner, often allowing for simpler operations and computations.
Common Misspellings for QUOTIENT RING
- 1uotient ring
- 2uotient ring
- wuotient ring
- auotient ring
- qyotient ring
- qhotient ring
- qjotient ring
- qiotient ring
- q8otient ring
- q7otient ring
- quitient ring
- quktient ring
- qultient ring
- quptient ring
- qu0tient ring
- qu9tient ring
- quorient ring
- quofient ring
- quogient ring
- quoyient ring
Etymology of QUOTIENT RING
The word "quotient" comes from the Latin term "quotiens", which means "how many times". It originally referred to the result of dividing one number by another. The word "ring" in this context refers to a mathematical structure also known as a "commutative ring".
The term "quotient ring" emerged in the field of abstract algebra, more specifically in ring theory, to describe a ring obtained from another ring by considering a congruence relation on the elements of the original ring and forming the equivalence classes, or cosets, of this relation. The resulting set of cosets, together with the operations inherited from the original ring, forms a new ring, known as the quotient ring.
Overall, the term "quotient ring" is a combination of the mathematical concepts of "quotient" and "ring", reflecting its origin and purpose in abstract algebra.
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