An integral element over a ring is an element in an extension ring that satisfies a polynomial equation with coefficients from the original ring. This concept is crucial for understanding the relationship between rings and their extensions, especially when examining properties like integral closure and integrally closed domains.
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An element 'a' in a ring extension R over a ring A is integral over A if there exists a monic polynomial $$f(x) = x^n + a_{n-1}x^{n-1} + ... + a_0$$ with coefficients from A such that $$f(a) = 0$$.
The set of all integral elements over a ring A forms a subring called the integral closure of A in R.
If R is a finitely generated A-algebra, then every element of R can be expressed as an integral element over A in some cases.
Integral elements are important in algebraic geometry as they correspond to affine varieties that are defined by polynomial equations.
An integral domain that contains all its integral elements is integrally closed, which implies that it has desirable algebraic properties, such as being normal.
Review Questions
What does it mean for an element to be integral over a ring, and how does this property relate to polynomial equations?
For an element to be integral over a ring means it satisfies a polynomial equation with coefficients from that ring. Specifically, an element 'a' is integral over a ring A if there exists a monic polynomial with coefficients in A such that when 'a' is substituted into this polynomial, the result is zero. This relationship emphasizes how algebraic properties of rings can influence the structure of their extensions.
Discuss the significance of integral extensions in the context of algebraic structures and provide an example.
Integral extensions are significant because they maintain many properties of the original ring while extending its structure. For example, if A is a Noetherian ring and R is an integral extension of A, then R inherits Noetherian properties. An example would be the ring of integers Z and its extension formed by adding the square root of 2; each element can be shown to satisfy polynomials with integer coefficients.
Evaluate the implications of being integrally closed for a domain, particularly in relation to its geometric interpretation.
Being integrally closed means that every element integral over a domain is contained within that domain, which has strong implications for both algebra and geometry. For instance, if a domain corresponds to a geometric object like an affine variety, being integrally closed ensures that all points defined by polynomial relations lie within the variety itself. This property enhances our understanding of how algebraic and geometric structures interact and informs us about the regularity and singularity characteristics of varieties.
A ring extension is integral if every element of the extension is integral over the base ring, meaning it satisfies a monic polynomial with coefficients from the base ring.
Integrally Closed Domain: A domain is integrally closed if every element that is integral over it belongs to the domain itself, ensuring that the domain contains all of its integral elements.
Monic Polynomial: A polynomial is monic if its leading coefficient is 1, which simplifies the analysis of integral elements as they satisfy such polynomials.