An integro-differential ring is a differential ring that is closed under an integration operation satisfying the fundamental theorem of calculus. Via the Newton--Leibniz formula, a generalized evaluation is defined in terms of integration and differentiation. The induced evaluation is not necessarily multiplicative, which allows to model functions with singularities and leads to generalized shuffle relations. In general, not every element of a differential ring has an antiderivative in the same ring. Starting from a commutative differential ring and a direct decomposition into integrable and non-integrable elements, we construct the free integro-differential ring. This integro-differential closure contains all nested integrals over elements of the original differential ring. We exhibit the relations satisfied by generalized evaluations of products of nested integrals. Investigating these relations of constants, we characterize in terms of Lyndon words certain evaluations of products that determine all others. We also analyze the relation of the free integro-differential ring with the shuffle algebra. To preserve integrals in the original differential ring for computations in its integro-differential closure, we introduce the notion of quasi-integro-differential rings and give two adapted constructions of the integro-differential closure. Finally, in a given integro-differential ring, we consider the internal integro-differential closure of a differential subring and identify it as quotient of the free integro-differential ring by certain constants.

翻译：积分-微分环是一种微分环，其在满足微积分基本定理的积分运算下封闭。通过牛顿-莱布尼茨公式，利用积分和微分定义了广义赋值。诱导的赋值不一定是乘性的，这使得能够建模具有奇点的函数并导出广义洗牌关系。一般而言，并非微分环中的每个元素都在同一环中存在原函数。从交换微分环出发，通过将其元素分解为可积与不可积部分的直和，我们构造了自由积分-微分环。该积分-微分闭包包含原始微分环中元素的所有嵌套积分。我们展示了嵌套积分乘积的广义赋值所满足的关系。通过研究这些常数关系，我们利用林登词刻画了决定所有其他赋值的特定乘积赋值。我们还分析了自由积分-微分环与洗牌代数的关联。为在积分-微分闭包的计算中保持原始微分环的积分结构，我们引入了拟积分-微分环的概念，并给出两种适配的积分-微分闭包构造方法。最后，在给定积分-微分环中，我们考虑微分子环的内部积分-微分闭包，并将其识别为自由积分-微分环对特定常数商的商环。