References & terminology.

An algebraic structure is called pseudosimple if it has more than one element and is isomorphic to all its nontrivial quotients. This concept was introduced in the paper

Monk, Donald
On pseudo-simple universal algebras.
Proc. Amer. Math. Soc. 13 (1962), 543-546.

The main result of the paper is that if $A$ is pseudosimple algebra, then the congruence lattice of $A$ is a well-ordered chain of order type $\omega^{\beta}+1$ for some ordinal $\beta$, and, conversely, for every ordinal $\beta$ there exists a pseudosimple lattice whose congruence lattice is a well-ordered chain of order type $\omega^{\beta}+1$.

A later paper that is closer to the problem on this page is:

Schein, Boris M.
Pseudosimple commutative semigroups.
Monatsh. Math. 91 (1981), no. 1, 77-78.

The main result here is that a commutative semigroup is pseudosimple iff it is a $2$-element semigroup, $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$ for some prime $p$. This paper states as a corollary to the main result that an abelian group is pseudosimple iff it is isomorphic to $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$

References & terminology.

An algebraic structure is called pseudosimple if it has more than one element and is isomorphic to all its nontrivial quotients. This concept was introduced in the paper

Monk, Donald
On pseudo-simple universal algebras.
Proc. Amer. Math. Soc. 13 (1962), 543-546.

The main result of the paper is that if $A$ is pseudosimple algebra, then the congruence lattice of $A$ is a well-ordered chain of order type $\omega^{\beta}+1$ for some ordinal $\beta$, and, conversely, for every ordinal $\beta$ there exists a pseudosimple lattice whose congruence lattice is a well-ordered chain of order type $\omega^{\beta}+1$.

A later paper that is closer to the problem on this page is:

Schein, Boris M.
Pseudosimple commutative semigroups.
Monatsh. Math. 91 (1981), no. 1, 77-78.

The main result here is that a commutative semigroup is pseudosimple iff it is a $2$-element commutative semigroup, $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$ for some prime $p$. This paper states as a corollary to the main result that an abelian group is pseudosimple iff it is isomorphic to $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$

References & terminology.

An algebraic structure is called pseudosimple if it has more than one element and is isomorphic to all its nontrivial quotients. This concept was introduced in the paper

Monk, Donald
On pseudo-simple universal algebras.
Proc. Amer. Math. Soc. 13 (1962), 543-546.

The main result of the paper is that if $A$ is pseudosimple algebra, then the congruence lattice of $A$ is a well-ordered chain of order type $\omega^{\beta}+1$ for some ordinal $\beta$, and, conversely, for every ordinal $\beta$ there exists a pseudosimple algebra whose congruence lattice is a well-ordered chain of order type $\omega^{\beta}+1$.

A later paper that is closer to the problem on this page is:

Schein, Boris M.
Pseudosimple commutative semigroups.
Monatsh. Math. 91 (1981), no. 1, 77-78.

The main result here is that a commutative semigroup is pseudosimple iff it is a $2$-element semigroup, $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$ for some prime $p$. This paper states as a corollary to the main result that an abelian group is pseudosimple iff it is isomorphic to $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$

References & terminology.

An algebraic structure is called pseudosimple if it has more than one element and is isomorphic to all its nontrivial quotients. This concept was introduced in the paper

Monk, Donald
On pseudo-simple universal algebras.
Proc. Amer. Math. Soc. 13 (1962), 543-546.

The main result of the paper is that if $A$ is pseudosimple algebra, then the congruence lattice of $A$ is a well-ordered chain of order type $\omega^{\beta}+1$ for some ordinal $\beta$, and, conversely, for every ordinal $\beta$ there exists a pseudosimple lattice whose congruence lattice is a well-ordered chain of order type $\omega^{\beta}+1$.

A later paper that is closer to the problem on this page is:

Schein, Boris M.
Pseudosimple commutative semigroups.
Monatsh. Math. 91 (1981), no. 1, 77-78.

The main result here is that a commutative semigroup is pseudosimple iff it is a $2$-element semigroup, $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$ for some prime $p$. This paper states as a corollary to the main result that an abelian group is pseudosimple iff it is isomorphic to $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$