For a finite involutive non-degenerate solution
(
X
,
r
)
(X,r)
of the Yang–Baxter equation it is known that the structure monoid
M
(
X
,
r
)
M(X,r)
is a monoid of I-type, and the structure algebra
K
[
M
(
X
,
r
)
]
K[M(X,r)]
over a field
K
K
shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand–Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid
M
(
X
,
r
)
M(X,r)
and the algebra
K
[
M
(
X
,
r
)
]
K[M(X,r)]
is much more complicated than in the involutive case, we provide some deep insights.
In this general context, using a realization of Lebed and Vendramin of
M
(
X
,
r
)
M(X,r)
as a regular submonoid in the semidirect product
A
(
X
,
r
)
⋊
Sym
(
X
)
A(X,r)\rtimes \operatorname {Sym} (X)
, where
A
(
X
,
r
)
A(X,r)
is the structure monoid of the rack solution associated to
(
X
,
r
)
(X,r)
, we prove that
K
[
M
(
X
,
r
)
]
K[M(X,r)]
is a finite module over a central affine subalgebra. In particular,
K
[
M
(
X
,
r
)
]
K[M(X,r)]
is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension bounded by
|
X
|
|X|
. We also characterize, in ring-theoretical terms of
K
[
M
(
X
,
r
)
]
K[M(X,r)]
, when
(
X
,
r
)
(X,r)
is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of
M
(
X
,
r
)
M(X,r)
.
These results allow us to control the prime spectrum of the algebra
K
[
M
(
X
,
r
)
]
K[M(X,r)]
and to describe the Jacobson radical and prime radical of
K
[
M
(
X
,
r
)
]
K[M(X,r)]
. Finally, we give a matrix-type representation of the algebra
K
[
M
(
X
,
r
)
]
/
P
K[M(X,r)]/P
for each prime ideal
P
P
of
K
[
M
(
X
,
r
)
]
K[M(X,r)]
. As a consequence, we show that if
K
[
M
(
X
,
r
)
]
K[M(X,r)]
is semiprime, then there exist finitely many finitely generated abelian-by-finite groups,
G
1
,
…
,
G
m
G_1,\dotsc ,G_m
, each being the group of quotients of a cancellative subsemigroup of
M
(
X
,
r
)
M(X,r)
such that the algebra
K
[
M
(
X
,
r
)
]
K[M(X,r)]
embeds into
M
v
1
(
K
[
G
1
]
)
×
⋯
×
M
v
m
(
K
[
G
m
]
)
\operatorname {M}_{v_1}(K[G_1])\times \dotsb \times \operatorname {M}_{v_m}(K[G_m])
, a direct product of matrix algebras.