Question

3.20. Prove that each of the following rings is not a field: (a) \(\mathbb{Z}[i]\). (b) \(\mathbb{R}[x]\). (c) \(\mathbb{H}\). (d) \(M_n(\mathbb{R})\) if \(n \ge 2\). 3.21. Let \(R\) be a commutative ring. Prove Proposition 3.15, which says that \(R\) is an integral domain if and only if it has the cancellation property.

          3.20. Prove that each of the following rings is not a field:
(a) \(\mathbb{Z}[i]\).
(b) \(\mathbb{R}[x]\).
(c) \(\mathbb{H}\).
(d) \(M_n(\mathbb{R})\) if \(n \ge 2\).
3.21. Let \(R\) be a commutative ring. Prove Proposition 3.15, which says that \(R\) is an integral domain if and only if it has the cancellation property.

3.20. Prove that each of the following rings is not a field:
(a) ℤ[i].
(b) ℝ[x].
(c) ℍ.
(d) Mn(ℝ) if n ≥ 2.
3.21. Let R be a commutative ring. Prove Proposition 3.15, which says that R is an integral domain if and only if it has the cancellation property.

Added by Alvaro C.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Madhur L

06/20/2023

Step 1

Step 1: To prove that aZ[i] is not a ring, we need to show that it does not satisfy the ring properties. Show more…

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3.20. Prove that each of the following rings is not a ring (aZ[i], bR, cH, dMnRif, n2). 3.21. Let R be a commutative ring. Prove Proposition 3.15, which says that R is an integral domain if and only if it has the cancellation property. R is an integral domain and it has finitely many elements.