diff --git a/docs/src/CommutativeAlgebra/affine_algebras.md b/docs/src/CommutativeAlgebra/affine_algebras.md
index b47e7d4aab26..bd82251e6376 100644
--- a/docs/src/CommutativeAlgebra/affine_algebras.md
+++ b/docs/src/CommutativeAlgebra/affine_algebras.md
@@ -357,13 +357,14 @@ julia> C1, (s,t) = graded_polynomial_ring(QQ, ["s", "t"]);
 julia> V1 = [s^3, s^2*t, s*t^2, t^3];
 
 julia> para = hom(D1, C1, V1)
-Map with following data
-Domain:
-=======
-Graded multivariate polynomial ring in 4 variables over QQ
-Codomain:
-=========
-Graded multivariate polynomial ring in 2 variables over QQ
+Ring homomorphism
+  from graded multivariate polynomial ring in 4 variables over QQ
+  to graded multivariate polynomial ring in 2 variables over QQ
+defined by
+  w -> s^3
+  x -> s^2*t
+  y -> s*t^2
+  z -> t^3
 
 julia> twistedCubic = kernel(para)
 ideal(-x*z + y^2, -w*z + x*y, -w*y + x^2)
@@ -375,13 +376,13 @@ julia> D2, (a, b, c) = graded_polynomial_ring(QQ, ["a", "b", "c"]);
 julia> V2 = [p2(w-y), p2(x), p2(z)];
 
 julia> proj = hom(D2, C2, V2)
-Map with following data
-Domain:
-=======
-Graded multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Quotient of multivariate polynomial ring by ideal with 3 generators
+Ring homomorphism
+  from graded multivariate polynomial ring in 3 variables over QQ
+  to quotient of multivariate polynomial ring by ideal with 3 generators
+defined by
+  a -> w - y
+  b -> x
+  c -> z
 
 julia> nodalCubic = kernel(proj)
 ideal(-a^2*c + b^3 - 2*b^2*c + b*c^2)
@@ -396,13 +397,13 @@ julia> C3, x = polynomial_ring(QQ, "x" => 1:3);
 julia> V3 = [x[1]*x[2], x[1]*x[3], x[2]*x[3]];
 
 julia> F3 = hom(D3, C3, V3)
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Multivariate polynomial ring in 3 variables over QQ
+Ring homomorphism
+  from multivariate polynomial ring in 3 variables over QQ
+  to multivariate polynomial ring in 3 variables over QQ
+defined by
+  y[1] -> x[1]*x[2]
+  y[2] -> x[1]*x[3]
+  y[3] -> x[2]*x[3]
 
 julia> sphere = ideal(C3, [x[1]^3 + x[2]^3  + x[3]^3 - 1])
 ideal(x[1]^3 + x[2]^3 + x[3]^3 - 1)
@@ -434,13 +435,13 @@ julia> C, p = quo(S, ideal(S, [c-b^3]));
 julia> V = [p(2*a + b^6), p(7*b - a^2), p(c^2)];
 
 julia> F = hom(D, C, V)
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Quotient of multivariate polynomial ring by ideal with 1 generator
+Ring homomorphism
+  from multivariate polynomial ring in 3 variables over QQ
+  to quotient of multivariate polynomial ring by ideal with 1 generator
+defined by
+  x -> 2*a + c^2
+  y -> -a^2 + 7*b
+  z -> c^2
 
 julia> is_surjective(F)
 true
@@ -462,13 +463,13 @@ julia> C, (s, t) = polynomial_ring(QQ, ["s", "t"]);
 julia> V = [s*t, t, s^2];
 
 julia> paraWhitneyUmbrella = hom(R, C, V)
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Multivariate polynomial ring in 2 variables over QQ
+Ring homomorphism
+  from multivariate polynomial ring in 3 variables over QQ
+  to multivariate polynomial ring in 2 variables over QQ
+defined by
+  x -> s*t
+  y -> t
+  z -> s^2
 
 julia> D, _ = quo(R, kernel(paraWhitneyUmbrella));
 
@@ -517,22 +518,22 @@ julia> L[1]
  -5*y + z
 
 julia> L[2]
-Map with following data
-Domain:
-=======
-Quotient of multivariate polynomial ring by ideal with 2 generators
-Codomain:
-=========
-Quotient of multivariate polynomial ring by ideal with 2 generators
+Ring homomorphism
+  from quotient of multivariate polynomial ring by ideal with 2 generators
+  to quotient of multivariate polynomial ring by ideal with 2 generators
+defined by
+  x -> x
+  y -> 2*x + y
+  z -> 10*x + 5*y + z
 
 julia> L[3]
-Map with following data
-Domain:
-=======
-Quotient of multivariate polynomial ring by ideal with 2 generators
-Codomain:
-=========
-Quotient of multivariate polynomial ring by ideal with 2 generators
+Ring homomorphism
+  from quotient of multivariate polynomial ring by ideal with 2 generators
+  to quotient of multivariate polynomial ring by ideal with 2 generators
+defined by
+  x -> x
+  y -> -2*x + y
+  z -> -5*y + z
 
 ```
 ## Normalization
diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Attributes.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Attributes.jl
index 45cb7c67e588..c8705dc86fde 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Attributes.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Attributes.jl
@@ -135,13 +135,13 @@ Spectrum
     by ideal(x1)
 
 julia> pullback(inclusion_morphism(X, Y))
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Quotient of multivariate polynomial ring by ideal with 1 generator
+Ring homomorphism
+  from multivariate polynomial ring in 3 variables over QQ
+  to quotient of multivariate polynomial ring by ideal with 1 generator
+defined by
+  x1 -> 0
+  x2 -> x2
+  x3 -> x3
 ```
 """
 pullback(f::AbsSpecMor) = pullback(underlying_morphism(f))
diff --git a/src/InvariantTheory/affine_algebra.jl b/src/InvariantTheory/affine_algebra.jl
index b8ede1a775fc..415d7b9b43e4 100644
--- a/src/InvariantTheory/affine_algebra.jl
+++ b/src/InvariantTheory/affine_algebra.jl
@@ -66,13 +66,7 @@ with generators
 AbstractAlgebra.Generic.MatSpaceElem{nf_elem}[[0 0 1; 1 0 0; 0 1 0], [1 0 0; 0 a 0; 0 0 -a-1]]
 
 julia> affine_algebra(IR)
-(Quotient of multivariate polynomial ring by ideal with 1 generator, Map with following data
-Domain:
-=======
-Quotient of multivariate polynomial ring by ideal with 1 generator
-Codomain:
-=========
-Graded multivariate polynomial ring in 3 variables over cyclotomic field of order 3)
+(Quotient of multivariate polynomial ring by ideal with 1 generator, Hom: quotient of multivariate polynomial ring -> graded multivariate polynomial ring)
 ```
 """
 function affine_algebra(IR::InvRing; algo_gens::Symbol = :default, algo_rels::Symbol = :groebner_basis)
diff --git a/src/Rings/MPolyMap/AffineAlgebras.jl b/src/Rings/MPolyMap/AffineAlgebras.jl
index 2efdc2fc9f3b..0f7e20021796 100644
--- a/src/Rings/MPolyMap/AffineAlgebras.jl
+++ b/src/Rings/MPolyMap/AffineAlgebras.jl
@@ -155,13 +155,11 @@ julia> is_bijective(F)
 true
 
 julia> G = inverse(F)
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 1 variable over QQ
-Codomain:
-=========
-D
+Ring homomorphism
+  from multivariate polynomial ring in 1 variable over QQ
+  to quotient of multivariate polynomial ring by ideal with 2 generators
+defined by
+  t -> x
 
 julia> G(t)
 x
diff --git a/src/Rings/MPolyMap/MPolyAnyMap.jl b/src/Rings/MPolyMap/MPolyAnyMap.jl
index 0eeef010584c..5654dd66d82f 100644
--- a/src/Rings/MPolyMap/MPolyAnyMap.jl
+++ b/src/Rings/MPolyMap/MPolyAnyMap.jl
@@ -80,10 +80,39 @@ _images(f::MPolyAnyMap) = f.img_gens
 #  String I/O
 #
 ################################################################################
+function Base.show(io::IO, ::MIME"text/plain", f::MPolyAnyMap)
+  io = pretty(io)
+  println(io, "Ring homomorphism")  # at least one new line is needed
+  println(io, Indent(),  "from ", Lowercase(), domain(f))
+  println(io, "to ", Lowercase(), codomain(f))
+  println(io, Dedent(), "defined by", Indent())
+  R = domain(f)
+  g = gens(R)
+  for i in 1:(ngens(R)-1)
+    println(io, g[i], " -> ", f(g[i]))
+  end
+  print(io, g[end], " -> ", f(g[end]), Dedent())
+  # the last print statement must not add a new line
+  phi = coefficient_map(f)
+  if !(phi isa Nothing)
+    println(io)
+    println(io, "with map on coefficients")
+    print(io, Indent(), phi, Dedent())
+  end
+end
 
-# There is some default printing for maps.
-# We might want to hijack this if we want to indicate whether there is
-# a non-trivial coefficient ring map.
+function Base.show(io::IO, f::MPolyAnyMap)
+  io = pretty(io)
+  if get(io, :supercompact, false)
+    # no nested printing
+    print(io, "Ring homomorphism")
+  else
+    # nested printing allowed, preferably supercompact
+    print(io, "Hom: ")
+    print(IOContext(io, :supercompact => true), Lowercase(), domain(f), " -> ")
+    print(IOContext(io, :supercompact => true), Lowercase(), codomain(f))
+  end
+end
 
 ################################################################################
 #
diff --git a/src/Rings/MPolyMap/MPolyQuo.jl b/src/Rings/MPolyMap/MPolyQuo.jl
index ca68f59bd288..6e1c86072449 100644
--- a/src/Rings/MPolyMap/MPolyQuo.jl
+++ b/src/Rings/MPolyMap/MPolyQuo.jl
@@ -66,13 +66,13 @@ julia> A, _ = quo(R, ideal(R, [y-x^2, z-x^3]));
 julia> S, (s, t) = polynomial_ring(QQ, ["s", "t"]);
 
 julia> F = hom(A, S, [s, s^2, s^3])
-Map with following data
-Domain:
-=======
-A
-Codomain:
-=========
-Multivariate polynomial ring in 2 variables over QQ
+Ring homomorphism
+  from quotient of multivariate polynomial ring by ideal with 2 generators
+  to multivariate polynomial ring in 2 variables over QQ
+defined by
+  x -> s
+  y -> s^2
+  z -> s^3
 ```
 """
 function hom(R::MPolyQuoRing, S::NCRing, coeff_map, images::Vector; check::Bool = true)
diff --git a/src/Rings/MPolyMap/MPolyRing.jl b/src/Rings/MPolyMap/MPolyRing.jl
index 4dc367163c7d..2764f81fe723 100644
--- a/src/Rings/MPolyMap/MPolyRing.jl
+++ b/src/Rings/MPolyMap/MPolyRing.jl
@@ -52,13 +52,14 @@ julia> K, a = FiniteField(2, 2, "a");
 julia> R, (x, y) = polynomial_ring(K, ["x", "y"]);
 
 julia> F = hom(R, R, z -> z^2, [y, x])
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 2 variables over GF(2^2)
-Codomain:
-=========
-Multivariate polynomial ring in 2 variables over GF(2^2)
+Ring homomorphism
+  from multivariate polynomial ring in 2 variables over GF(2^2)
+  to multivariate polynomial ring in 2 variables over GF(2^2)
+defined by
+  x -> y
+  y -> x
+with map on coefficients
+#1
 
 julia> F(a * y)
 (a + 1)*x
@@ -69,13 +70,20 @@ julia> Qi, i = quadratic_field(-1)
 julia> S, (x, y) = polynomial_ring(Qi, ["x", "y"]);
 
 julia> G = hom(S, S, hom(Qi, Qi, -i), [x^2, y^2])
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 2 variables over imaginary quadratic field defined by x^2 + 1
-Codomain:
-=========
-Multivariate polynomial ring in 2 variables over imaginary quadratic field defined by x^2 + 1
+Ring homomorphism
+  from multivariate polynomial ring in 2 variables over imaginary quadratic field defined by x^2 + 1
+  to multivariate polynomial ring in 2 variables over imaginary quadratic field defined by x^2 + 1
+defined by
+  x -> x^2
+  y -> y^2
+with map on coefficients
+  Map with following data
+    Domain:
+    =======
+    Qi
+  Codomain:
+    =========
+    Qi
 
 julia> G(x+i*y)
 x^2 - sqrt(-1)*y^2
@@ -87,13 +95,12 @@ julia> f = 3*x^2+2*x+1;
 julia> S, (x, y) = polynomial_ring(GF(2), ["x", "y"]);
 
 julia> H = hom(R, S, gens(S))
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 2 variables over ZZ
-Codomain:
-=========
-Multivariate polynomial ring in 2 variables over GF(2)
+Ring homomorphism
+  from multivariate polynomial ring in 2 variables over ZZ
+  to multivariate polynomial ring in 2 variables over GF(2)
+defined by
+  x -> x
+  y -> y
 
 julia> H(f)
 x^2 + 1
diff --git a/src/Rings/localization_interface.jl b/src/Rings/localization_interface.jl
index 64fff1f21052..1969a2543510 100644
--- a/src/Rings/localization_interface.jl
+++ b/src/Rings/localization_interface.jl
@@ -173,13 +173,13 @@ Localization
   at complement of prime ideal(x)
 
 julia> iota
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Localization of multivariate polynomial ring in 3 variables over QQ at complement of prime ideal
+Ring homomorphism
+  from multivariate polynomial ring in 3 variables over QQ
+  to localization of multivariate polynomial ring in 3 variables over QQ at complement of prime ideal
+defined by
+  x -> x
+  y -> y
+  z -> z
 ```
 """
 function Localization(S::AbsMultSet)
diff --git a/src/Rings/mpoly-affine-algebras.jl b/src/Rings/mpoly-affine-algebras.jl
index 941f99e9665b..a1dd415629ab 100644
--- a/src/Rings/mpoly-affine-algebras.jl
+++ b/src/Rings/mpoly-affine-algebras.jl
@@ -1267,13 +1267,12 @@ Quotient
   by ideal(-T(1)*y + x, -T(1)*x + y^2, T(1)^2 - y, -x^2 + y^3)
 
 julia> LL[1][2]
-Map with following data
-Domain:
-=======
-A
-Codomain:
-=========
-Quotient of multivariate polynomial ring by ideal with 4 generators
+Ring homomorphism
+  from quotient of multivariate polynomial ring by ideal with 1 generator
+  to quotient of multivariate polynomial ring by ideal with 4 generators
+defined by
+  x -> x
+  y -> y
 
 julia> LL[1][3]
 (y, ideal(x, y))
diff --git a/src/Rings/mpoly-localizations.jl b/src/Rings/mpoly-localizations.jl
index 5e582bd90acb..64223ab988a0 100644
--- a/src/Rings/mpoly-localizations.jl
+++ b/src/Rings/mpoly-localizations.jl
@@ -1168,13 +1168,13 @@ Localization
   at complement of prime ideal(x)
 
 julia> iota
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Localization of multivariate polynomial ring in 3 variables over QQ at complement of prime ideal
+Ring homomorphism
+  from multivariate polynomial ring in 3 variables over QQ
+  to localization of multivariate polynomial ring in 3 variables over QQ at complement of prime ideal
+defined by
+  x -> x
+  y -> y
+  z -> z
 ```
 """ localization(R::MPolyRing, U::AbsMPolyMultSet)
 
@@ -2877,22 +2877,20 @@ julia> PHI = hom(RQL, TL, TL.([t, t^2, t^3]));
 julia> PSI = hom(TL, RQL, RQL.([x]));
 
 julia> phi = restricted_map(PHI)
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Localization of multivariate polynomial ring in 1 variable over QQ at complement of maximal ideal
+Ring homomorphism
+  from multivariate polynomial ring in 3 variables over QQ
+  to localization of multivariate polynomial ring in 1 variable over QQ at complement of maximal ideal
+defined by
+  x -> t
+  y -> t^2
+  z -> t^3
 
 julia> psi = restricted_map(PSI)
-Map with following data
-Domain:
-=======
-Multivariate polynomial ring in 1 variable over QQ
-Codomain:
-=========
-Localization of quotient of multivariate polynomial ring at complement of maximal ideal
+Ring homomorphism
+  from multivariate polynomial ring in 1 variable over QQ
+  to localization of quotient of multivariate polynomial ring at complement of maximal ideal
+defined by
+  t -> x
 ```
 """
 restricted_map(PHI::MPolyLocalizedRingHom) = PHI.res