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Prove if a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is two-sided and is also the unique maximal right ideal in $R$.

Two definitions of Jacobson Radical

If $A=k[x_1,\dots,x_n]/I$ is a finitely generated $k$-algebra with $A\cong k^m$ as modules, what can we say about $I$?

For which $d \in \mathbb{Z}$ is $\mathbb{Z}[\sqrt{d}]$ a unique factorization domain?

Bezout in $\mathbb C [x,y]$ [duplicate]

Maximal ideal not containing the set of powers of an element is prime

Subring which is not an ideal?

Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups. [duplicate]

How to determine whether a unique factorization domain is a principal ideal domain?

R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$ [duplicate]

Intuitive understanding of ideal $I = (x+1,x^2+1)$ and the quotient $\Bbb Z[x]/I$

the image of $1$ by a homomorphism between unitary rings

Question about ideals and linear polynomials

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$M$ maximal iff $\bar{M}$ is maximal

Prove if a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is two-sided and is also the unique maximal right ideal in $R$.

Two definitions of Jacobson Radical

If $A=k[x_1,\dots,x_n]/I$ is a finitely generated $k$-algebra with $A\cong k^m$ as modules, what can we say about $I$?

For which $d \in \mathbb{Z}$ is $\mathbb{Z}[\sqrt{d}]$ a unique factorization domain?

Bezout in $\mathbb C [x,y]$ [duplicate]

Maximal ideal not containing the set of powers of an element is prime

Subring which is not an ideal?

Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups. [duplicate]

How to determine whether a unique factorization domain is a principal ideal domain?

R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$ [duplicate]

Intuitive understanding of ideal $I = (x+1,x^2+1)$ and the quotient $\Bbb Z[x]/I$

the image of $1$ by a homomorphism between unitary rings

Question about ideals and linear polynomials