Extension and contraction of I to D^-1R

In summary, the definition of the extension and contraction of ideals can be found at the bottom of page 708 in Dummit and Foote (Chapter 15, Section 15.4 Localization). The notation for these symbols is similar to I^e and I^c, with the superscripts e and c occurring before the I. To display these symbols in latex, use the codes ^eI for ^eI and ^cI for ^cI. For better spacing and alignment, consider using StackExchange for options.
  • #1
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At the bottom of page 708 in Dummit and Foote (Chapter 15, Section 15.4 Localization) we find the definition of the extension and contraction of ideals.

The notation is similar to \(\displaystyle I^e \) and \(\displaystyle I^c \) except that the superscripts e and c occur before the I.

Can someone please help me with the latex for these symbols?

Peter
 
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  • #2
Use the codes:

^eI to get: \(\displaystyle ^eI\)

^cI to get: \(\displaystyle ^cI\)
 
  • #3
MarkFL said:
Use the codes:

^eI to get: \(\displaystyle ^eI\)

^cI to get: \(\displaystyle ^cI\)
Thanks Mark

Appreciate the help.

Peter
 
  • #4
See also StackExchange for some options that fine-tune the spacing.
 
  • #5
One thing I've noticed in latex is that if you put an "empty character" (using {}) before a preceding superscript/subscript, it seems to give better alignment with other similarly tagged symbols.
 

FAQ: Extension and contraction of I to D^-1R

What is the meaning of "extension and contraction" in I to D^-1R?

The terms "extension" and "contraction" refer to the process of extending or contracting a mathematical structure known as the "Idealizer" (I) to the "Double Reciprocal" (D^-1R). This process is commonly used in algebraic geometry to study algebraic varieties and their properties.

How is the extension and contraction of I to D^-1R used in scientific research?

This process is used in various areas of scientific research, such as in the study of dynamical systems, differential equations, and control theory. It allows researchers to analyze the behavior and stability of complex systems by transforming them into simpler and more manageable forms.

What is the relationship between I and D^-1R in the process of extension and contraction?

I and D^-1R are two mathematical structures that are closely related in the process of extension and contraction. I is an idealizer, which is a set of elements that can be multiplied with any element of a ring (a mathematical structure with addition and multiplication operations) to produce another element within the ring. D^-1R is the double reciprocal of a ring, which is obtained by taking the inverse of each element in the ring and then applying the same multiplication operation. The process of extension and contraction involves transforming I into D^-1R and vice versa.

What are some practical applications of the extension and contraction of I to D^-1R?

The extension and contraction of I to D^-1R has many practical applications in fields such as computer science, physics, and engineering. It is used in the design and analysis of control systems, signal processing, and data compression algorithms. It is also used in the study of geometric structures, such as manifolds and surfaces, and in the development of numerical methods for solving differential equations.

Can you provide an example of the extension and contraction of I to D^-1R?

One example of the extension and contraction of I to D^-1R is the process of converting a linear differential equation into an algebraic equation. This is done by replacing the derivatives in the differential equation with elements from the idealizer and then transforming them into the double reciprocal form. This allows for easier analysis and solution of the equation.

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