Algebra-Rings, Subrings and Ideals, Homomorphisms of Rings [IAS (Admin.) Mains Mathematics]: Questions 1 - 5 of 5

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Question 1

Appeared in Year: 2015

Describe in Detail Subjective▾

Give an example of a ring having identity but a subring of this having a different identity. (CS main Paper 2)

Edit

Explanation

Let and S =

First, we prove R…

… (779 more equations) …

Question 2

Appeared in Year: 2011

Describe in Detail Subjective▾

Let F be the set of all real valued continuous functions defined on the closed interval prove that is a commutative ring with unity with respect to addition and multiplication of function defined point wise as below:

Where (Paper II)

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Explanation

  • Given

    T. P…

… (370 more equations) …

Question 3

Appeared in Year: 2013

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Let of all real valued continuous functions on , under the operations

Let

Is M a maximal ideal of R? Justify your Answer. (Paper II)

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Explanation

  • Define

    by

  • Let

… (217 more equations) …

Question 4

Appeared in Year: 2014

Describe in Detail Subjective▾

Prove that the set is a commutative ring with identity. (Paper II)

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Explanation

  • Given
  • I. is Abelian group

    (i) Let

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Question 5

Appeared in Year: 2007

Describe in Detail Subjective▾

Let where . Show that R is a ring under matrix addition and multiplication

Then show that A is a left ideal of R but not a right ideal of R. (Paper II)

Edit

Explanation

  • First, we show that R is a ring under matrix addition and multiplication

    I. Is an abelian group

  • (a) Closure property:

    Let

… (653 more equations) …