Kruse says that Paul speaks of five aspects of God’s righteousness in Roman. These are all aspects of the same umbrella idea: ‘God acting in accordance with his own nature for the sake of his name’:-

1. God’s distributive justice. This is implied in Rom 1:18-2:11, where Paul says that God deals with all people in accordance with his revelation, on the one hand, and in accordance with their works, on the other hand. See also Rom 3:1-20
2. God’s covenant faithfulness. In Rom 3:3-9 Paul says that God’s judgment of Israel is not due to any failure on his part to remain faithful to his covenant. It is, rather, due to Israel’s sin. See also Rom 9:1-29; 11:1-10. In the latter passage, Paul argues that God, in his covenant faithfulness, has always kept a remnant of Israel.
3. God’s saving action. According to Rom 3:21-26 God manifests his righteousness in providing for redemption through the death of Christ, enabling by faith a righteousness (right standing) before him.
4. God’s gift of justification and restoration of a right relationship with himself. This is already anticipated in Rom 3:21-26, and expounded with reference to Abraham in Rom 4. See also Rom 5:17 (‘the gift of righteousness’) and Rom 9:30-10:4.
5. God’s gift of righteousness which leads to righteous living in believers. In Rom 6:1-23 Paul teaches that those who are under grace (i.e. who are justified) are no longer slaves of sin, but are slaves of righteousness.

Kruse (along with Moo and others) thinks that in Rom 1:17 ‘righteousness’ should be construed as referring to God’s saving action whereby he brings people into a right relationship with himself.

K. L. Onesti and M. T. Brauch (DPL, art. ‘Righteousness, Righteousness of God’) insist that ‘the concept of God’s righteousness, its nature, function and result, is central to Paul’s teaching on the justification of the sinner. The genitive construction dikaiosynē theou, “righteousness of God” (Rom 1:17; 3:5, 21, 22; 10:3; 2 Cor 5:21), or dikaiosynē autou, “his righteousness” (Rom 3:25, 26), or hē ek theou dikaiosynē, “righteousness from God” (Phil 3:9) are found ten times. Most of these are located in Romans, Paul’s fullest discourse on God’s redemptive work in Christ.’

Romans 1:16–17 is foundational for understanding the meaning of this concept.’  In the gospel ‘“God’s righteousness is revealed.” In fact, ‘gospel’ and ‘God’s righteousness’ are ‘virtually synonymous’.  ‘In all three passages in Romans (Rom 1; 3; 10), as well as in Philippians 3:9, the apprehension of God’s righteousness and of its coming into the sphere of human experience is related to faith in Jesus and understood as a gift of God’s grace. In addition, all the texts explicitly include a rejection of “works of the Law” as a valid instrument for attaining or receiving this righteousness of, or from, God.’

Historically (according to Onesti and Brauch), the concept of God’s righteousness has been understood in Greek and Latin categories, ‘where righteousness as a quality of God’s character is either given to us and makes us righteous, or is the basis for God’s judical pronouncement, declaring us righteous.’

Recent scholarship, on the other hand, has recognised the OT background for Paul’s teaching.  ‘The idea of God’s righteousness appears prominently in salvation texts, where God’s redemptive action toward his covenant people is defined by this term. It is God’s righteousness which saves from enemies, from threatening situations, from the state of alienation from God. In such settings God’s righteousness is frequently defined by the terms “steadfast love” and “faithfulness” (e.g., Is 11:5; 16:5; Ps 5:7–8; 89:13–14; 98:2–3). These relational attributes are in some contexts virtually synonymous with “righteousness” and “salvation” (e.g., Ps 85:7–13). Thus, God’s righteousness may be rendered as “saving deed” or “relation-restoring love.”’  The ‘righteousness of God’, then, does not refer to a divine attribute, but rather to ‘God’s forgiving love and redemptive intervention in the world through Christ.’