Integral dependence refers to a relationship between elements in a ring extension where an element is integral over another if it satisfies a monic polynomial with coefficients from that ring. This concept is crucial for understanding how algebraic structures behave under extensions, highlighting the interplay between elements and the rings they belong to.
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An element 'b' in a ring extension R/S is integral over S if there exists a monic polynomial with coefficients in S such that 'b' is a root of that polynomial.
If every element of an extension ring R is integral over a base ring S, then R is called an integral extension of S.
Integral dependence is transitive; if an element 'c' is integral over 'b', and 'b' is integral over 'a', then 'c' is also integral over 'a'.
The integral closure of a ring in a larger ring consists of all elements that are integral over the original ring, serving as an important concept in commutative algebra.
Integral dependence ensures that certain properties, such as Noetherian properties, are preserved when moving between rings.
Review Questions
How does the concept of integral dependence relate to the notion of integral elements within a ring extension?
Integral dependence indicates that an element can be described through a polynomial relationship with coefficients from its base ring. When we say an element is integral, it means it fulfills this condition, being expressed as a root of such a monic polynomial. Thus, every integral element shows a specific form of dependence on the base ring, illustrating how elements interact and form structures in ring extensions.
Discuss how transitivity plays a role in understanding integral dependence among multiple elements in different rings.
Transitivity in integral dependence allows us to connect multiple elements across different rings by showing that if one element depends on another, and that second element depends on yet another, then we can conclude that the first element also depends on the third. This property simplifies the analysis of complex relationships in ring extensions and helps establish broader principles about how rings and their elements interact when considering integrality.
Evaluate the implications of integral extensions for the structure of algebraic systems and their properties.
Integral extensions have significant implications for algebraic structures, particularly because they preserve many desirable properties such as Noetherian conditions. When an extension is integral, it often retains characteristics like being finitely generated or retaining ideal properties, which are crucial for further analysis and application in algebraic geometry and number theory. Understanding these implications helps mathematicians use integral extensions as tools for solving problems related to algebraic varieties and rings.
Related terms
Integral element: An element of a ring extension that satisfies a monic polynomial with coefficients in the base ring.
Integral extension: A ring extension in which every element of the extended ring is integral over the base ring.
Monic polynomial: A polynomial whose leading coefficient is equal to 1.